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MINIMUM MEAN-SQUARED ERROR ADAPTIVE ANTENNA ARRAYS FOR 
DIRECT-SEQUENCE CODE-DIVISION MULTIPLE ACCESS SYSTEMS 



By 
JOHN EARLE MILLER 



A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL 

OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULRLLMENT 

OF THE REQUIREMENTS FOR THE DEGREE OF 

DOCTOR OF PHILOSOPHY 



UNIVERSITY OF FLORIDA 
1996 



UNIVERSITY OF FLORIDA LIBRARES 



Copyright 1996 

by 
John Earle Miller 



This work is dedicated to my wife Kim, my children Marissa and Christopher and my 
parents Frank and Garnet. 



ACKNOWLEDGMENTS 

I would like to thank all of the members of my committee for their involvement, 
guidance and influence in this research. I would like to extend a special thanks to my 
chairman. Dr. Scott L. Miller, for his support, many insights and thoughtful suggestions 
which have influenced this work. The financial support provided by the Department of 
Electrical and Computer Engineering is also gratefully acknowledged. 



IV 



TABLE OF CONTENTS 

page 

ACKNOWLEGMENTS iv 

ABSTEIACT vii 

CHAPTERS 

1 INTRODUCTION 1 

Direct-Sequence Spread-Spectrum with Multiple- Access 3 

Adaptive Antenna Arrays 7 

Previous Work 13 

Dissertation Outline 21 

2 A MULTIUSER ANTENNA ARRAY PROCESSOR 
STEADY-STATE PERFORMANCE 24 

The Multiuser MMSE Processor 25 

Spatially Orthogonal Users 26 

Nonorthogonal Users 30 

Single-User System 34 

One Weak User, One Strong User 34 

Two Equal-Power Users 36 

Two Strong Users 36 

Numerical Results 37 

3 THE MULTIUSER ARRAY PROCESSOR 

ADAPTIVE PERFORMANCE 42 

Mean-Based Performance Measure 43 

Variance-Based Performance Measure 47 

Simulation Results 49 

4 BASE STATION RECEIVER PERFORMANCE USING A 
SMART ANTENNA ARRAY IN A DS-CDMA SYSTEM 

WITH IMPERFECT POWER CONTROL 52 



System Description 53 

Analysis 54 

Simulations and Results 61 

Conclusions 65 

5 BASE STATION RECEIVER PERFORMANCE WITH A 
SMART ANTENNA ARRAY IN THE PRESENCE OF 
MULTIPATH FADING AND SHADOW FADING 68 

A Single Cell with Signals Subjected to Rayleigh Fading 69 

Multiple Cells with Signals Subjected to Rayleigh Fading 

and Shadow Fading 7j 

Analysis 76 

Results gQ 

Conclusions/Discussion gg 

Summary g^ 

6 BASE STATION ANTENNA ARRAY 

ADAPTIVE PERFORMANCE 9I 

Signal and Channel Model 92 

Adaptive Receivers 94 

Simulations 9-7 

Conclusions/Summary jq2 

7 SUMMARY 106 

Areas for Future Work 109 

APPENDIX 

A THE OUTPUT SNR PERFORMANCE SURFACE 1 13 

LIST OF REFERENCES II7 

BIOGRAPHICAL SKETCH I29 



vi 



Abstract of Dissertation Presented to the Graduate School 
of the University of Florida in Partial Fulfillment of the 
Requirements for the Degree of Doctor of Philosophy 



MINIMUM MEAN-SQUARED ERROR ADAPTIVE ANTENNA ARRAYS FOR 
DIRECT-SEQUENCE CODE-DIVISION MULTIPLE- ACCESS SYSTEMS 

By 

John Earle Miller 

August, 1996 

Chairman: Dr. Scott L. Miller 

Major Department: Electrical and Computer Engineering 

This dissertation examines the performance of direct-sequence code-division multiple- 
access receivers which use a minimum mean-squared error adaptive antenna array as a 
predetection spatial filter. The array attenuates multi-access interference prior to 
conventional, direct-sequence matched-filter detection. Conventional detectors are 
vulnerable to heightened levels of multi-access interference. Minimum mean-squared error 
processing seems like a natural choice for optimization in a code-division multiple-access 
environment since several efficient search algorithms exist which are compatible with 
decision-directed equalizer structures. 

Two array structures are examined. The first suoicture uses a single set of array 
weights to equalize more than one desired signal. For two strong incident signals of 
unequal power levels, the steady-state response gives output signal-to-noise ratios that are 



vu 



leveled to a value near the maximum response of the weaker of the two users. The steady- 
state adaptive performance of the multi-user array based on a least-mean-squares 
algorithm is also examined. It is found that the maximum allowable step size based on 
stabiUty arguments will also give adequate output signal-to-noise ratio performance of the 
multi-user processor. 

The second structure uses a single set of weights per multiple-access signal and the 
array output feeds a conventional detector. The array/detector performance measure is 
outage probability and outage-based capacity as a function of the number of array 
elements and the degree of power control error. A robust, incremental measure of 
performance~the per-element capacity-is defmed as the capacity per array element for a 
given outage probability. Steady-state performance is evaluated for the case of directional 
signals as well as for the case of signals subjected to multipath fading and shadowing. The 
adaptive performance of the recursive least-squares algorithm is also investigated. 



viii 



CHAPTER 1 
INTRODUCTION 

Wireless communication based on direct-sequence spread spectrum (DS-SS) has 
received considerable attention as an efficient signaUng format for code-division multiple- 
access systems (CDMA). While DS-CDMA has a long history of use in defense 
applications where jamming resistance and security are primary concerns,' some 
proponents maintain that it will also serve as a high-capacity format for the next- 
generation mobile-cellular and wireless systems for commercial use.^'^"*' System 
specifications for domestic use have been endorsed by industrial agencies,* the results of 
field trials have been pubUshed'* and it has been studied as a possible format for third- 
generation mobile radio systems in Europe.'-'" 

Despite a frenzy of activity and interest, DS-SS has a serious drawback: the near-far 
effect This occurs when strong signals overwhelm a weaker desired signal during the 
detection process. In a commercial mobile-ceUular system the near-far effect can occur at 
the ceU-site base-station receiver. Incident signals originating from multiple-access, fixed- 
power transmitters geographically distributed throughout a ceU can have incident power 
levels that change drasticaUy as the transmitter positions vary. Signals originating from 
transmitters near the base station may overwhehn received signals from transmitters on the 
fringe of the cell. Vigorous proponents of commercial DS-CDMA systems have developed 
transmit power-control formats to adjust each user's RF transmit power in real time so 
that power levels of all multiple-access signals incident upon the base station are 



approximately equal. Initially, it was stated that power control would need to be able to 
adjust the transmit power over a 80 dB dynamic range in a mobile ceUular scenario." Field 
tests indicate that 50 dB is more likely.^'^ 

Another use for DS-SS is in the Global Positioning System (GPS). Originally 
implemented as a positioning system for the Department of Defense, it has enjoyed 
commercial appUcations in mapping, navigation, and surveying." The world-wide GPS 
uses a DS-SS signaling format to obtain relatively accurate estimates of geographical 
position. An earth-bound GPS receiver determines its position by measuring the path 
delays of several DS-SS signals which originate from earth-orbiting satellites. The near-far 
effect is not a critical issue for many commercial GPS applications, such as aviation or 
shipping, since Uie signals are subjected only to free-space path losses. Some commercial 
appUcations, such as surveying, suffer other forms of signal losses, such as multipath 
fading or shadow fading, and these can indirectiy lead to near-far limited performance. 

This work examines receiver performance when an adaptive antenna array is used at 
tiie DS-SS receiver. In such a system Uie array would act as a front-end spatial filter which 
preserves the integrity of tiie desired signal or signals, attenuates interferere and 
supplements the existing power conti-ol algoritiim. Such an approach might be compatible 
with existing cellular CDMA standards. The remainder of tiiis chapter is divided into 
several sections. The next section wUl give a quaUtative review of DS-SS systems. The 
second section will review quantities and expressions used in the analysis of the minimum 
mean-squared-error (MMSE) beamforming antenna arrays. The Uiird section wUl review 
previous work germane to tiie research presented here and tiie last section will provide an 
outline of the remainder of this dissertation. 



Direct-Sequenc e Spread- Spectrum with Multiple-Access 

Generating a DS-SS waveform involves multiplying a modulated baseband waveform 
by a psuedorandom sequence sometimes known as a pseudonoise (PN) code. In an 
asynchronous multi-access channel the incident DS-SS waveforms may be subjected to 
random time delays and carrier phase delays and corrupted by noise. For a single-channel 
receiver the multiple-access incident signals may be modeled by the expression: 

Yit) = Relf^A„c„{t-r„)b„{t-T„)cxp{j{(o,t + d„)) + n{t)\ (1.1) 



lm=l 



where the index m refers to the mth of K signals. The quantity Am is the incident signal 
level of the mth signal, Cm(t-tm) represents the mth PN code waveform which consists of a 
sequence (of length Nc) of psuedorandom square pulses, (cm e {+1,-1 }) each with interval 
Tc The quantity b4t-Tm) represents independent binary phase-shift keyed (BPSK) 
modulation with equally-likely symbols (fc„ e {+1,-1 }) obtained via ideal, square pulses. 
The time delay Tm is uniformly distributed over a symbol interval [0,Tb). The time delay t„ 
arises because the channel allows asynchronous access; users may begin transmission at 
any time. The carrier frequency is designated by cot, and 0„ is the random carrier phase 
uniformly distributed over [0,27t). The quantity n(t) is complex additive white Gaussian 
noise (AWGN) with a one-sided power spectral density of No. 

Note that in contrast to systems which use frequency-division multiple access 
(FDMA), all users in a CDMA system share a common frequency cok. Frequency-division 
channels are achieved by separating the carrier frequencies of multiple, bandlimited signals 
to the point that they do not interfer with one another in the receiver. Code-division 



channels in CDMA systems, on the other hand, are achieved through the low 
crosscorrelation properties of the individual PN sequences. The PN sequences have many 
interesting properties, including low crosscorrelation''* defined as 

^» (^) = J c„ (f )c, {t + T)dt «Nc V m vt n. The limits of the integral are from zero to 
NcTc. In a CDMA system the individual users share a common carrier frequency but are 
assigned distinct PN sequences that permit code-division channeled Unks to a base station 
or other receiver. 

This woric assumes that detection of DS-SS signals is accomplished by a correlating 
detector also referred to as a conventional detector. The ith user's DS signal is multiplied 
by a synchronized replica of its PN sequence, passed to an integrate-and-dump filter and 




Figure 1.1 Conventional detector, baseband model. 



hard-limited to provide an estimate of the modulation symbol, as shown in Figure 1.1. 

The conventional detector provides optimum performance for a single user in AWGN 
but gives sub-optimum performance in a multi-access environment The performance of a 
conventional detector in a multi-access environment depends roughly on the processing 
gain which is a measure of the interference rejection capability of the detector and is 
sometimes defined as the number of code symbols (or "chips") per modulation symbol (Nc 
= Tb/Tc). For a fixed, finite processing gain it is always possible for strong interferers to 



overwhelm the detection of a weaker signal and cause poor performance. The complex 
baseband output of the integrate-and-dump portion of the conventional detector devoted 
to user 1 consists of a desired signal component as well as multiaccess intefercnce and 
noise components: 



So.i{t) = AMt)+Z,K \c,{t)c„{t-T„)b„{t-T„)dt+ \ci{t)nit)dt 



m=2 I 

K 



= AMt)+^A„[b„__,R„,{T„) + b„^,R„,{z„)\+ \c,it)n{t)dt 

Kl{'^n,)=Jc„{t-tJc,(t)dt 





(1.2) 



where: ° 



T- 



where bit 1 of user 1 is the detected signal of interest and T; = 0. The quantities R„ ^ (t„ ) 
and ^„ 1 (t„ ) are the partial crosscorrelation functions. The quantities bm,.j and bm.o 
represent the contributions of the mth user's two bits which overlap the current bit interval 
of the desired user. The values of crosscorrelations depend upon the sequences and their 
relative time delays. The multi-access bit-error performance of the conventional detector 
has been studied extensively; the dependence of the detector performance on PN-code 
parameters has been well characterized.''-'* If one or more crosscorrelations are non-zero 
and the corresponding input levels are large, those components may overwhelm the 
desired signal in the detector. This is the near-far effect. 

In mobile cellular or personal communication systems, signals incident on the base 
station (also referred to as tiie uplink signal patii) can take wide-ranging incident power 
levels because of the random placement of users about the base station. The current 



solution to this dilemma is to vary the transmitter power levels in real time via transmit 
power control. In a system using closed-loop power control the base station receiver 
monitors the detected power level of each multi-access signal. As the incident power 
levels of the multi-access signals vary because of changing channel conditions, instructions 
are sent from the base station to the mobile units, via an uncoded narrowband side 
channel, to individually increase or decrease transmit power. The aim of the power control 
is to force the effective bit-energy-to-noise special density ratio i(Eb/No)S of all the 
incident signals to the same threshold. In practice, however, the power control is unable to 
perfectly track changing channel conditions because of doppler-induced multipath fading 
and finite power control step size. Error in the power-controlled signal results in an 
effective bit energy to noise specu^ density ratio {(Eb/NoUff) at the detector output which 
is approximately lognormal in distribution.'^'* Field trials indicate that high-mobility 
mobUe users (such as in fast-moving automobUes) have (Et/NoUff standard deviations of o, 
= 2.5 dB while lower values (o; = 1.5 dB) have been recorded for low-mobUity users 
(pedestrians).' 

Existing cellular CDMA specifications also have a contingency for open-loop power 
control.^ In an open-loop power-controlled system the mobile or personal units use the 
detected base-station carrier power as a reference to adjust their transmit power. No side 
channel is required. Some researchers have proposed estimation algorithms which would 
allow a mobile to make estimates of the base station carrier power. The researchers 
present evidence that the carrier-measurement error in terrestrial mobile cellular systems is 
weU-approximated as lognormaUy distributed with a or,„ < 3 dB for vehicle speeds of up 
to 60 mph and 8 dB of shadow fading. '^ Other researchers working in the field of mobUe 



cellular systems based on low-earth-orbiting-satellltes (LEOS) have also found open-loop 
power control error to be well-approximated by a lognormally distributed random variable 
(r.v.).'"-'' 

If we generalize the previous work and assume that power control error may be 
approximated by a lognormal r.v., then equation (1.1) may be rewritten slightly: 

Yit) = Rei^ApO''-^c„{t-z,)b„{t-T„)exp(j{coj + e„)) + n{t)\ (1.3) 

where e„ is normally distributed with zero mean and variance equal to Oj,^^ (N(0, o^J )). 
Note that in previous works the lognormal approximation of power control error applied 
to the incident power levels for open-loop power control, and to the (Et/No)^ for the case 
of closed-loop power control. To simplify analysis, this research will assume that power 
control error results in lognormally distributed incident signal levels for closed-loop or 
open-loop power-controlled systems. 

Adaptive Antenna Arrays 

This research assumes an adaptive antenna array is used as a front-end spatial filter to 
attenuate direction-dependent cochannel interference prior to conventional detection of 
multi-access DS-CDMA signals. The anticipated role of the adaptive array wUl be to 
attenuate the strongest interferers and thereby lessen performance degradations due to the 
near-far effect. The antenna arrays explored here will be limited to beamforming arrays - 
linear combiners - which use a minimum mean-squared error (MMSE) optimization 
criterion. This technique is also referred to as Wiener filtering^^ or optimum combining." 
A MMSE optimization is well suited to a mobile DS-SS-CDMA scenario since each user's 



PN sequence, required for DS-SS detection and decoding, can provide a convenient 
reference waveform. There are efficient MMSE adaptive algorithms which would require 
no more side information than the code sequence and its timing.^^ 

The array is composed of ideal isotropic sensors with no mutual coupling between the 
elements. Incident signals are spatially sampled by the sensors as they propagate across the 
array. The incident signals are assumed to be narrowband. This approximation is valid for 
spread spectrum signals as long as signal bandwidths are a small percentage of the carrier 
frequency. Current specifications* call for bandwidths of 1.2288 MHz at a carrier 
frequency of 850 MHz. The resulting double-sided signal bandwidth is much less than one 
percent, so the narrowband approximation should be acceptable. 

The physical displacements between the array sensors induce relative phase shifts on 
our spatially sampled, narrowband signals. The phase-shifts are constant over frequency 
and depend upon the array geometry and the signal's direction-of-arrival (DOA). The 
complex signal component outputs of the A^ discrete antenna elements for the mth incident 
signal can be described by a vector: 



Sc^ = .?«(Oexp(y(/:y„ - (oj)) = ^„(Oexp(- jcoj) 
where: 



exp(y^„.o) 
exp(y0„,i) 

.exp(70„.^_,)_ 



(1.4) 



u„=[exp(;0„,) exp(y0„,) ... exp(y0„^_, )]"" 

where the vector y„ (see Figure (1.2)) represents the physical displacements between 
some reference point and the A^ array elements. An element of y^ (y„^n n = 0...N-1) 



represents the physical distance between a reference point and the corresponding array 
element The parameter k is the free-space wave number of the incident plane wave {k = 2 
k/X ). Carrier phase-shifts have been absorbed into the complex, baseband function 
gjt).lihe vector Un, contains electrical phase shifts resulting from the physical 
displacements yn, between the elements and the point of reference. The individual elements 
of Un, ( i.e. (j>m4, ) represent the mth signal's phase shift between the reference point and the 
nth array element. The vector Um will be referred to as the interelement phase-shift vector 
or simply the phase-shift vector and will not be considered a function of time unless stated 
otherwise. For the remainder of the report constants will appear as normal block or script 
characters, vectors will be represented by lower-case boldface and matrices by upper-case 
boldface. 

A diagram illustrating some of the physical quantities is shown in Figure 1.2. An 
incident signal propagates as a plane wave s(t).The signals and noise are spatiaUy sampled 
by each element of the array; ignoring an explicit time dependence allows the samples to 
be represented by x„, n = 0...N-1, and form the input vector x. The inputs are weighted 
(via the weight vector w = [wo w, ...wn.,]"^) and summed to form the beamfomer output. 
The quantity ky is the phase-shift of the propagating wave due to physical displacement y. 
As tiie wave propagates across tiie array tiie phase-shifts due to displacement give rise to 
interelement phase-shifts which are contained witiiin Uie vector u. The lower part of tiie 
figure illustrates how tiie direction-dependent phase-shift arises between two adjacent 
elements. The quantity (j) is tiie phase shift between any two adjacent elements. 

A change in array geometry would affect only tiie functional form of tiie relative 
phase-shifts between tiie array elements. The quantity = 2nd(sine)/X in Figure 1.2 above 



10 

is the direction-dependent phase-shift between any two adjacent array elements for a linear 
array. Arrays composed of individual elements arranged in a circle will be examined in 
later chapters because, unlike linear arrays, they can resolve incident signals over the 
direction-of-arrival interval [0, In). The phase-shift between any two adjacent elements is 
given by ^ = (7uicos[6-2ji/N])/(Xsin[7c/NJ) for a circular array. 



plane wave: 




For a propagating plane wave s(t) 



for a linear 
array : 



y = dsin0 

^ = ity = 2;r y / A 

d = element spacing 



the phase - shift (p 
between the two 
elements is then: 



^ = 



2;r dsin0 



Figure 1.2 A directional plane wave incident on a linear array of sensors. 



The baseband array output vector is Uie sum of Uie noise and signal vectors: 

^ = nW + Zs™{t) =n{t) + f^gjt)u^ (1.5) 



m=l 



m-1 



11 

where n(t) is the AWGN vector. The noise is spatially and temporally white. The 
narrowband portion of the array autocorrelation matrix may be expressed as a sum of 
outer products or as a product of matrices: 

^ = E[x'x^]=Gll + f^Alulul =all + V'A'V' (1.6) 

m-l 

where U is the matrix of phase-shift vectors and A is a diagonal matrix of input 
amplitudes, (•)' represents a complex conjugate and {•f a vector transpose, respectively. 
The expectation £[•] averages the AWGN, n(t) which is assumed to be N(0.a,^). The 
norm-squared of Um is: 

ikir=">™='^ (1.7) 

independent of signal DOA where (•)" represents the hermitian transpose. 

The MMSE weight vector minimizes the mean-squared error between the processor 
output and the reference signal. The weight vector is given by: 

w„ = R"*p (1.8) 

where p is the steering vector and is given by: 

P = E[r{t)s'j] = AjAgUj (1.9) 

and Sd and Ud are the signal vector and phase-shift vector of the desired signal respectively. 
The quantity r(t) is the time varying reference waveform with a peak amplitude of A/?. The 
MMSE weight vector maximizes the output signal-to-interference-and-noise-ratio" 
(SINK) which is defined as the quotient of the desired signal power and the sum of the 
interference and noise power SINE = Ps/(Pi+Pn). The output signal-to-noise ratio (SNR) 
for the mth signal is the quotient of the output power of the mth signal and the output 



12 



noise power. It is given by: 



27 



ijn rorn- I m 



|2 



SNRo„ = -^ = 5A^/?/„ ' ; ,;' < N ■ SNRi„ (1. 10) 



The quantity SNRi„ = A^Va^ is the incident SNR for the mth signal, Ps.m is the desired 
signal output power, P, is the interference output power and Pn is the output noise power. 
For a single signal in AWGN, Wo = u*, and the upper bound becomes an equality. The 
choice of Wo = u^* represents the spatial equivalent of a matched filter (to the rath signal) 
and is referred to as a conventional beamformer^'*, a classical beamformeT^\ or a 
maximal ratio combiner}^ 

The sum total of the number of beams and nulls an array is capable of directing 
simultaneously is N-1, one less than the number of array elements. This is also referred to 
as the degrees-of-freedon?^ of an array. 

At this point, it may be important to explain the difference between the terms 
adaptive array and diversity combiner. The term adaptive array implies that the processor 
uses an array of sensors and exploits the phase and amplitude shifts between the array 
elements, induced by directional signals, to make processing decisions. A diversity 
combiner, on the other hand, exploits some degree of statistical independence between the 
input samples to ensure signal integrity at the summed output For example, in the case of 
communication channels which contain multipath fading, an array of sensors might provide 
statistically independent spatial samples. If a signal is faded at one sensor, it might not be 
faded at anoUier sensor; the statistical independence of die inputs is exploited by die 
processor to improve tiie integrity of the overall output response. There are many diversity 
combining strategies.'' A popular strategy for analytical purposes, mentioned previously 



13 

as maximal-ratio combining, maximizes the output SNR but does not allow adaptive 
interference suppression. The diversity-combining counterpart to MMSE filtering is 
referred to as optimum combining. It is sometimes possible to achieve statistically 
independent samples through time sampUng. Aldiough not an issue in this research, a few 
examples of time diversity will be cited in the following section. 

Previous Work 

This section is a survey of previous work in adaptive beamforming antenna arrays and 
in CDMA that is pertinent to this research. The first part of the survey will give a 
historical perspective to research in adaptive antenna arrays. Details of the earlier work 
oriented towards radar and military communications is admittedly sparse. As the timeline 
and focus become more current the coverage will become more detailed. The second part 
of the survey will review the most recent work in which adaptive arrays function as an 
integral component of DS-CDMA systems. The last part of the survey wUl examine 
previous work in cellular CDMA ( without antenna arrays) as it appUes to this research. 

A beam-steered array was investigated by Applebaum^' but the results were not 
published in open literature until a decade later.^° Widrow et al.^' reported on an antenna 
array using a least-mean-squares (LMS) processor in 1967. A special issue of the IEEE 
Transactions on Antennas and Propagation devoted to active and adaptive antennas was 
pubUshed in 1964,'' 1976'' and 1986.^^ Other processors were investigated by Frost'', 
Griffiths'^ and Schorr." Much of the initial adaptive array work focused on the 
performance of particular processors or radar-oriented appUcations. One exception to this 
was the maximal ratio diversity combiner investigated by Brennan.'* 



14 

Subsequent work explored the role of adaptive arrays in communication systems.^*' '^^ 
In particular, Compton'^ presented a qualitative evaluation of an experimental adaptive 
array in a DS communication system. The evaluation focused on a single desired signal 
and a limited number of jammers. One of the conclusions reached by Compton is that the 
array makes appreciable contributions to interference suppression. Winters'" studied the 
acquisition performance of an LMS adaptive array in a DS system using four-phase 
modulation with two PN codes, a short code for rapid acquisition and a long code for 
protection against jammers. Ganz'*^ evaluated the bit-error-rate (BER) performance of a 
receiver employing an adaptive array and one of several detectors for binary-phase-shift- 
keying (BPSK), quadrature phase-shift keying (QPSK), or differential phase-shift keying 
(DPSK) modulation. The receiver was subjected to continuous-wave (CW) jamming. 
Several authors have investigated the use of antenna arrays for mobile or personal 
communications. Bogachev and Kiselev'*' evaluated optimum-combining diversity arrays 
for the case of a single interferer. Winters" conducted a comparative study of optimum 
combining and maximal ratio combining base station arrays in a multiuser mobile 
environment with multipath fading but no shadow fading. The results quantify the possible 
improvement in SINR if optimum combining is selected over maximal ratio combining. 
Like optimum combining, maximal ratio combining preserves die integrity of the desired 
signal, but unlike optimum combining maximal ratio combining has no abihty to adaptively 
suppress interference. Winters did propose the use of psuedonoise codes to generate the 
LMS reference, but transmit power control was menuoned only for its effect on the 
convergence properties of the LMS array. He did not investigate in any detail the 
condition when the number of users greatly exceeds the number of array elements. Winters 



15 

also explored the use of adaptive arrays on base stations for in-building systems using 
dynamic channel assignment'" He again considered a PN-coded PSK modulation and 
circumvented power control considerations by assuming interferers were of equal power 
and much stronger than the desired signal. A more recent study"' investigated the 
acquisition and tracking performance of LMS and sample-matrix-inverse (SMI) 
beamformers in a time-division multiple-access system. 

Yeh and Reudink"*^ examined the contributions made by spatial diversity combiners to 
specu-al efficiency in FDMA cellular systems when the arrays are located on the base 
station and on the mobiles. Glance and Greenstein"^ also examined the contributions of 
diversity order (die number of array elements) on average BER in a mobile FDMA system. 
Vaughan'^ discussed the benefit of MMSE combining at the mobUe in an FDMA system 
and concluded with the comment that for MMSE combining to be successful wide 
bandwidth signals, such as those found in spread-spectrum systems, are necessary. 

More recently, adaptive array research has examined commercially-oriented 
applications as CDMA and non-CDMA wireless systems gain popularity. At this time, 
antenna arrays are under investigation as a means of providing space-division channels in 
multiple-access systems. The technique, called space-division multiple access (SDMA) by 
some authors, utilizes the spatial filtering properties of tiie array to selectively receive 
signals which share the same time slot and tiie same frequency band. The SDMA 
technique might apply to eitiier time-division multiple-access systems (TDMA) or CDMA 
systems. Swales et al."' established tiiat a steerable, multi-beam antenna array can increase 
tiie capacity and spectral efficiency of a cellular system. Suard and Kailath'" studied tiie 
upper bound of Uie information-based capacity of a wireless system which used a base- 



16 

Station antenna array in the uplink path. Experimental studies have been conducted. Xu et 
al. and Lin et al.'^ have examined algorithms based on direction-finding techniques 
MUSIC^^ and ESPRIT.^ They have concluded that SOMA techniques based on DOA 
estimation will not be effective in multipath-rich environments. Xu and Li" developed an 
SDMAADMA protocol. Ward and Compton**" examined the contributions that a 
receiving array can make to the performance of an ALOHA system. 

Arrays which include spatial and temporal adaptive processing nested within LMS 
feedback loops were proposed by Kohno et al.'* and Ko et al.'' Kohno used an LMS array 
in conjunction with adaptive temporal filtering which successively canceled DS signals due 
to multi-access interference. Ko described a null-steering beamformer nested within an 
LMS loop, not necessarily restricted to CDMA applications. Both considered limited 
multi-access scenarios with deterministic interference parameters. 

Diversity combining is not necessarily restricted to space diversity. Balaban and 
Salz • examined in detail the performance of a general multi-channel MMSE combiner 
working in conjunction with a decision-feedback equalizer. They estabUshed an upper 
bound on BER which is a function of the MMSE. A great deal of work has also been 
devoted to temporal diversity combiners which consist of a single input to a bank of 
matched filters, the outputs of which are coherently combined via maximal ratio 
combining. This is the basis for the RAKE" receiver as well as variations studied by otiier 
autiiors. Several structures were examined by Turin^^ while Uhnert and Pursley" 
examined diversity combining in multiuser CDMA system in which successive bits are 
spread witii different PN code subsequences. Wang et al. consider diversity for an indoor 
DS-CDMA system with Rician fading.*' 



17 

Some authors have studied the possible contributions made by arrays to the upUnk 
path performance in cellular systems. Liberti and Rappaport have studied the effects of a 
directive, steered-beam base-station receiving array on uplink performance in a CDMA 
cellular system with perfect power control. The study focused on the effect of beam shape 
and beam width on the average BER. It was found that beam width has the greatest 
impact on performance and that adding a three-element array to the base station can 
improve BER performance by three orders of magnitude. Tsoulos, Beach and Swales have 
recently examined the role of adaptive antenna technology in large "umbrella" cells 
overiaying smaller microcells in cellular CDMA systems^ ; they also examined the outage- 
based capacity enhancement due to an adaptive antenna array using a recursive least- 
squares (RLS) processor in a multi-ceU CDMA environment^^ The latter study was 
confined to simulations of the total interference to calculate outage. The authors 
concluded that an antenna with 6 dB of directivity gain can increase capacity by a factor of 
five. 

Winters, Salz and Gitlin^ studied the effects of optimum combining spatial diversity 
arrays on the capacity of a TDMA system in which the total number of incident signals 
was less than or equal to the array DOE. They applied previous work^-^' which resulted in 
an upper bound on BER, The assumption of high input SNR allowed a zero-forcing 
approximation to the optimum combining solution. Using these assumptions, and 
examining analytical expressions for BER, they found that optimum combining with N 
antennas and K interferers gives the same results as a maximal ratio combiner with N-K+1 
elements and no interferers. Their theoretical results, as they point out, no longer apply 
when the number of interferers exceeds the number of antenna elements. 



18 



A structure which combines the spatial filtering of an antenna array and the temporal, 
diversity-combining properties of a RAKE receiver have been proposed and investigated 
by Khalaj in concert with several other authors.^"'"'^' The structure is intended for use in 
channels with frequency-selective multipath fading. The structure allows the resolution of 
identifiable multipath rays by the time-filtering properties of the RAKE combiner and by 
the spatial filtering properties of the antenna array. 

A number of authors have conducted brief simulation-based studies of adaptive arrays 
for DS-CDMA systems. Yoshino et al." examine the simulation performance of two 
RLS-based spatial diversity combiners operating in concert with (Viterbi) sequence 
estimators. One processor subtracts estimates of the cochannel interference from the 
output prior to estimation of the desired user's data sequence while the other processor 
does not. Wang and Cruz" examine the pattern behavior and BER of a six-element arrays 
based on the RLS and ESPRIT algorithms with six active users with well-separated 
DOAs. Liu^" examined the performance of an LMS array with a scenario-dependent 
matrix preprocessor which aids in interference cancellation. Hanna et al." investigated the 
BER performance of a two-element LMS array which operates in conjunction with an 
adaptive equalizer. 

Perhaps the most in-depth study of the possible contributions of MMSE adapuve 
beamforming arrays to the performance of mobUe ceUular CDMA systems using closed- 
loop power control has been made by Naguib in concert with other authors. Initial work^* 
focused on steady-state performance of an array of sensors, each foUowed by a DS 
conventional detector, which functions as a post-detection combiner (termed code- 
filtering by the authors). Analysis resulted in a simple expression for capacity. The signal 



19 

model was for unfaded signals originating in an isolated single cell. Power control error 
was modeled by assigning a single interferer an incident signal level 10 dB higher than the 
other signals. 

Naguib, Paulraj and Kailath" extended the steady-state model in order to determine 
the outage probability in a cellular system with BPSK modulation, perfect power control, 
shadow fading, multipath fading, and cochannel interference equivalent to two tiers of 
surrounding cells. Assuming that the array pattern response consisted of a main lobe and 
no sidelobes resulted in a simple, closed-form expression for an upper bound on outage 
probability which simplified to the single-channel results of Gilhousen et al." when the 
array is reduced to a single element. Modeling the MMSE processor as a maximal ratio 
processor and assuming the interference was Gaussian resulted in a simpler expression for 
the outage probability upper bound.^* Naguib and Pauh^j then modified the simulation 
model to include DPSK-modulated signals and determined the Eriang capacity." The 
uplink performance widi M-ary orthogonal modulation was examined as well. *° 

A recursive beamforming algorithm was proposed by Naguib and Paukaj*' and 
simulated results were presented. The algorithm performed recursive updates on the 
matrix square root of the inverse of the covariance matrix. The authors claimed that the 
accumulation of numerical and quantization errors may cause the covariance matrix 
inverse to cease being hermitian defmite and that updating the matrix square root will 
allow the covariance matrix inverse to remain hermitian definite even when the matrix 
square root is not. Thus, say the authors, numerical instabUities are avoided. 

Naguib and Paulraj*^ continued their invesUgation of cellular base station antenna 
arrays by examining the tradeoffs in coverage area, mobile transmit power, and capacity 



20 

that are available when an array is used and the users are subjected to perfect power 
control. Using simplifying assumptions the authors derived expressions which generaUzed 
the effect of the antenna array on performance. This research will attempt to extend some 
of the results to the case of imperfect power control. 

Later investigations have resulted in detailed, simulation-based studies of an IS-95 
system which uses orthogonal signaling, forward error-correction coding, closed-loop 
power control and a base-station antenna array. Unlike their previous studies Naguib and 
Pauhaj appUed a model for imperfect closed-loop power control. They studied the 
standard deviation of power control error, although where the error is defined is not clear 
in the paper. Using simulation results they show that power control error dependence on 
die power control step size, the number of array elements, and the maximum doppler 
frequency and the spread in DOA of the multipath rays.*^ The dependence of BER on the 
same parameters was the topic of a subsequent paper.^ 

A multiuser LMS array for use in GPS receivers was examined by Beach et al.*' in a 
simulation study. The results were confined to plots which show die evolution of the 
adaptive array pattern over time in the presence of CW jammers; no steady-state results 
were presented. The near/far effect was not an issue in the study. 

A variety of authors have investigated the performance of cellular CDMA systems 
with imperfect power control and no antenna array. Simpsora and Holtzman** used 
simplified analytical models in order to provide insight into the interactions between power 
control, coding and interleaving. Stuber and Kchao'^ examined a multiple-cell CDMA 
system and evaluated the dependence of BER as a function of Uie distance from die base 
station. Jalali and Mermelstein** conducted a simulation study of a microcellular CDMA 



21 

system which included imperfect closed-loop power control and antenna diversity with 
square-law combining. Milstein et al/' studied the average BER performance of a 
multiple-cell system in which users where subjected to power control error which was 
described by a uniform random variable. Newson and Heath'" examined a CDMA system 
which suffered from imperfect sectoring and lognormally-distributed power control error 
and made capacity comparisons to TDMA and FDMA systems. 

Dissertation Outline 

The remainder of the dissertation will be divided into six chapters. Chapter 2 will 
examine some aspects of the steady-state performance of a multiuser MMSE array; some 
limited aspects of an adaptive LMS version were investigated by Beach et al.*^ The 
near/far performance will first be investigated for strong, spatially orthogonal users and 
will then be extended to the case of two users with any DOA spacing. The analysis will 
focus on the ability of the array to confine the signal outputs to the same output SNR. The 
results will show that the array may be suitable when signals are well-separated in DOA 
and do not outoumber the array DOFs. The analysis will borrow some of the analytical 
techniques used by Gupta'' and apply them to evaluate the steady-state performance of 
die multi-user MMSE array. Unlike either previous study, however, the analysis will focus 
on the performance of the array in a near/far environment. 

The third chapter will examine the adaptive performance of the multiuser MMSE 
array when it is implemented as an LMS processor. Analysis will show that the rule-of- 
thumb for picking step size to ensure stability also serves as a limit to reduce the spread in 
output signal levels due to LMS misadjustment 



22 

The fourth chapter will examine the steady-state performance of an MMSE array 
feeding a conventional detector. Unlike the array of the first two chapters, this array will 
be dedicated to a single user among many multiple-access users. The single-cell users will 
not be subjected to many of the influences normally found in a mobile environment, such 
as multipath fading, shadow fading or interference which originates from surrounding 
cells. This will simpUfy the analysis: the intent is not to provide an analysis fraught with 
mathematical rigor, but to use simplifying assumptions to aid in the derivation of closed- 
form analytical expressions which accurately predict the uplink performance and which 
clearly show the dependence of uplink capacity on the system parameters, especially the 
array and the degree of power control error. The performance measures will be outage 
probability and outage-based capacity. 

The analysis of chapter four will result in a simple closed-form expression for the 
capacity which is linear in the product of processing gain and the number of array elements 
and which decays exponentially with increasing power control error. The slope of the 
capacity Une (the per-element capacity) will serve as a robust measure of the array's 
incremental contribution to capacity. Although some of the assumptions are highly 
idealized (i.e. no multipath fading), diey will allow some weU-understood quantities, such 
as array gain, to be exploited in the analysis. Some of the work in this chapter is related to 
that of Padovani^ and also to Naguib and Paub-aj.*^ The details wiU be discussed during the 
chapter's derivations and discussions. 

Chapter 5 will extend the steady-state multi-access model of Chapter 4 to include the 
effects of multipath fading, shadow fading and outer-cell interference. The performance 
measures will again be the steady-state outage probability and outage-based capacity. 



23 

The sixth chapter will examine the recursive-least-squares (RLS) adaptive 
performance of the MMSE adaptive array in a mobile cellular scenario in a nonstationary 
channel. The discrete-event simulation model includes time-varying multipath fading, 
stationary shadow fading and time-varying outer-cell interference. 

The last chapter will give a brief summary and some conclusions of this research. It 
will also identify some areas of future work. 

This research makes contributions in two areas. First, analysis and numerical solutions 
provide more insight into the steady-state and the adaptive behavior of the multi-user 
MMSE array (chapters 2,3) than was provided by previous authors. Analysis shows that 
the array can remove die near-far effect and level the output SNRs to nearly the same 
level, under certain circumstances. Second, the analysis and simulation of a single-user 
array operating in conjunction with a conventional detector results in simple expressions 
for capacity when the signal sources are subjected to imperfect power control (chapters 
4,5,6), The results characterize the incremental contributions an adaptive array can make 
to uplink performance. This is in contrast to previous works which examine the effects of 
closed-loop power control error only in simulation for Umited scenarios. 



i«' 



CHAPTER 2 

A MULTIUSER ANTENNA ARRAY PROCESSOR 

STEADY-STATE PERFORMANCE 

This chapter examines the ideal, steady-state performance of the multiuser MMSE 
array proposed by Beach et al.*^ The authors conducted a simulation study of the pattern 
behavior of a multiuser LMS adaptive array with and without directional interference. 
They did not provide any analytical results which give general insight into the array's 
steady-state performance. In related work, Gupta" examined a multi-user steered-beam 
array for non-CDMA applications and simplified his analysis by assuming the special case 
of spatially orthogonal users. Gupta was interested in using the array's effective aperature 
to improve the output SNR of the desired signals and in using the adaptive properties to 
reject interference. Other authors have shown that steered-beam arrays and LMS arrays 
give die same output SNR performance as long as the steering vectors differ by a 
constant.'^ Using diis rationale it might seem that Gupta's work could provide some 
insight to this problem. However, he was trying to configure the array processor to avoid 
the power leveUng effect that this work is attempting to exploit. 

This chapter will focus on the array output SNR performance for strong incident 
signals. The analysis will deemphasize the role of the spread-spectrum signalling other 
than to note that it provides a relatively easy way to provide separation and detection of 
the multiple, desired signals present at the single array output. The remainder of this 
chapter will be divided into several sections. The first section will provide some qualitative 
information about the multiuser MMSE array processor and give some refinements to the 

24 



25 

general array equations given in the introduction. The second section will develop 
analytical expressions of the output SNR for K <N-1 spatially orthogonal users. The third 
section will present analytical expressions for two users which have arbitrary DO As and 
are therefore not necessarily spatially orthogonal. The last section will compare the 
analytical and the numerical results and discuss their implications. 

The Multiuser MMSE Processor 

The multiuser array treats each of the K signals as a component of a single composite 
desired signal. This is accomplished via a reference signal which is a sum of the modulated 
PN sequences of the desired incident signals. A block diagram is shown below in Figure 
2.1. In the figure all functions of time are ignored. For analytical simplicity we will assume 
that perfect estimates of the modulation and code waveforms are available for the 
generation of the reference waveform. This will not usually be the case. Generation of the 
reference signal can be a challenging issue and it has been investigated by other 
authors."-^-"' 

The array will force the outputs of the individual signals to the levels of that signal's 
component of the reference waveform. To avoid the near/far effect the individual 
components comprising the reference waveform have the same peak ampUtudes. Since all 
of the incident signals share the same beamformer weights, the output SNRs will be 
leveled to approximately the same value. The analysis will show that the array will force 
the output SNRs to values very close to the maximum output SNR of the weakest user 
imder certain conditions. 



26 



The steering vector defined by equation (1.9) represents the crosscorrelation between 
the reference signal and the multiple, independent input signals. It becomes a sum of 
single-user steering vectors: 



P = SP™=/?XA„"m 



(2.1) 



where A„ and Um are the incident amplitude and phase-shift vector of the mth signal, 
respectively. 



V 



Xo(t) 



o 



o 




Output 




XN-l(t) 



Wo 



o 
o 
o 



<> 



Reference 
signal 



e(t) 



Wn-I 





To a bank of 
conventional 
DS detectOTs 



noit) + ^So„{t) 



B=l 



^.0 - T,>,0- T.) 
I 



^)^b,(t- T,>,0- T,) 



Figure 2.1 Multiuser MMSE processor 



Spatially Orthogonal Users 

Consider the limiting case of multiple users which are spatially orthogonal to one 
another (uj^i^ =0 V i^tk ). The minimum source separation (in DOA) which aUows 
spatial ortiiogonality corresponds to tiie Rayleigh limit.'' The Rayleigh limit is generally 



27 



considered to be the minimum amount of source separation which still allows resolution of 
two sources by a beamformer.^'* If the users are orthogonal and the array elements are 
isotropic the inverse of the autocorrelation matrix is:'' 



R-' = 



. ^ NSNRi„ . T 

1-2. - "- »- 



/IN. m- 



Til+NSNRi, 



(2.2) 



and the resulting MMSE weight vector, Wo = R p, is given by: 



w. 



=-i- 

^l ;^i+ 



A„ 



NSNRL 



•u. 



(2.3) 



Substituting these quantities into the general expression for the n'th users output SNR 
(equation (1.10)) gives: 



SNRo. = 



NSNRi„ 
l + NSNRi 



\2 



f K 



■nj 



NSNRL 



^-^ 



^„til + N-SNRiJ% 



(2.4) 



Some coarse judgements on performance for multiple users can be made using 
equation (2.4). If {SNRim » 1 V m =1 ... K} the output SNRs for all of the users are 
approximately equal to: 



r K 



SNRo'^ N- 



* 1 
y^iSNRi„ 



(2.5) 



If there are K users with the same input SNR = SNRi then equation (2.5) simplifies to: 

NSNRi 



SNRo = 



K 



(2.6) 



which shows that die users will all have output SNRs equal to the maximum possible SNR 
divided by die number of active users. If die array parameters remain constant, die 
performance will degrade as the number of users increases. 



If there are S users with the same input SNRi = kjSNR and K-S users with SNRi = 
SNR then the output SNR for the K users smiplifies to: 

SNRo = N ■ SNR ^ (2 1) 

If kj-(K- S) » S (a possible near/far scenario) the expression for ou^ut SNR 
simplifies further: 

i N-SNR 



SNRo 



K ^ S 

LNSNR (2.8) 

' K = S 



K 



where: 

S = no. of strong users with input SNRi = kj SNR, ki » S 
K = total number of users 
J^ - 5 = no. of weaker users with input SNRi = SNR 

From the equation immediately above it is apparent that one weak user {K-S = 1) will 
dominate the overall performance of the array. This might make intuitive sense, even for 
the general case. A minimum MSE processor wiU try to force each output signal to have 
the same value as that signal's component of the reference waveform. This strategy will 
favor weaker incident signals which need large weight values to cause their array output to 
match their component of the reference waveform ( i.e. minimum error power). Large 
incident signals will be attenuated via phase shifts and the noise power - the denominator 
of equation (2.4) - will be large thus causing a lower than maximum output SNR even for 
strong input signals. 

Equation (2.8) above establishes that the weakest incident signal drasticaUy affects die 
overall performance of the array. The worst array response towards an arbitrary signal 
(SNRon) for a different input signal (SNRifn) can be found by finding the value of SNRi^ 



29 



which forces dSNROfJdSNRif^, the derivative of the nth signal's output SNR with respect 
to the mth signal's input SNR, to zero: 



dSNRo^ 
dSNRL 



NSNRi„ 
\\ + N-SNRi„ 



r, XT rxro. \ /" D 



l-NSNRL 



l + N-SNRi 



m J 



NSNRi, 



\-2 



fiil + NSNRi,)^ 



= 



(2.9) 



From the equation above its is evident that a minimum occurs for SNRi^^i = 1/N. 
Substituting SNRifn = 1/N into equation (2.4) gives (for SNRi^ » l,n^m): 



SNRo, =4- 



SNRo^ = 1 



^ NSNRi„ Y 
^l + N-SNRi^^ 



-4 



(2.10) 



For SNRifn = 1/N the value of tiie mth eigenvalue of R is twice tiiat of tiie noise-only 

eigenvalues. At this input level the array is barely able to resolve the m'th input signal 
from the input noise. 

It might be useful to define an output SNR "spread" which bounds die output SNRs 
for aU of tiie users. From equation (2.4) it can be seen tiiat tiie largest and smallest input 
signals result in the largest and smallest output SNRs respectively. We can predict tiie 
output spread by tiie difference between tiie largest and smallest output SNRs: 



^SNRo = SNRo^^ - SNRo_ 



^ NSNRL., V f NSNRU V 



{l + N-SNRi„ 



{l + N-SNRi„ 



^ NSNRi„ Y 
ha + N-SNRij'j 



(2.11) 



If we let SNRimax pass to infinity the output spread becomes: 



lim 



30 



1 ( NSNRi^^ Y 
[l + N-SNRU 



^ NSNRij 



(2.12) 



The equation above simplifies to: 



(2.13) 



ASNRo = ^_±y^lSNRi^ 

An upper bound on the output SNR spread results: 

ASNRo < 2 + i (2 U) 

NSNRi^^ 

miD 

In summary it would appear that for N-SNRim » 1, m=l...D, the user outputs will 
be near the value of the maximum output SNR for the weakest user and the output SNR 
spread will be equivalent to 2. 

Nonorthogonal TTsp.rs 

Analyzing the general case of multiple narrowband users which are not spatiaUy 
ortiiogonal is difficult because of the matrix inverse in equation (1.8). When tiie users are 
not orthogonal the analytical matrix inverse (via the matrix inversion lemma) quickly 
becomes intractable as the number of users increases beyond unity. The general case of 
only two users was examined in detail. The approach taken here is to derive expressions 
for the two output SNRs. The expressions are tiien interpreted as transfer functions with 
poles and zeros which are dependent upon Uie input levels and tiie DOAs. Critical points 
of the functions are then evaluated. 



31 



The autocorrelation matrix R for two independent, narrowband users in AWGN 
(equation (1.6)), is given by: 



= o^[l+ SNRiyy, + SNRiyy^] 

The inverse of R (R-1) may be calculated via die matiix inversion lemma: 
D-i _ 1 T 1 SNRii . T 



(2.15) 



(j; \ + SNRi, 



'i"i 



SNRL 



' 1 + NSNRL " 



(2.16) 



u]u\SNRL 
l + NSNRL ' 



I + SNRL- 



T • 



SNRL SNRL 



1 + SNRi^ 
The steering vector p which results from equation (2.1) is: 



P = /2A,u; + ^u; = (J„R^SNRiy, + a^R^fsMI^u, (2. 17) 

Substituting equations (2.16) and (2.17) into equation (1.8), wo = R'^p, gives tiie MMSE 
weight vector: 

"'"^^'^^'^o [^/^^(l + S^^h ) - uJulSNRi, 4SNRi;]a[ + 

R fi , , (2.18) 

— ySNRi, (1 + SNRi, ) - u]u[SNRi, V^AWTju; 

The quantity D^ is defined by: 

A =l + N(SNRi, +SNRi,) + [N-\uJul\yNRi,SNRi, (2.19) 

Two vector inner products are required to find the output SNR defined in equation 
(1. 10). If we assume tiie point of reference for the array is die same as the physical center 
of the array, Uie imier products will be real.^^ An inner product between two phase-shift 



32 

vectors may then be expressed in dot-product form: uj"u* =Ncosa^^ where a^ is the 

angle between the two vectors in signal space. Substituting this relationship into equation 
(2.18): 

II ||2 R^ , . . 

Ko i = -^ ^[SNRi, + SNRi^ ) + 2N^SNRi,SNRi^ cosa,^ + 

n o 

2 



R 

^^N'SNRi.SNRi, sin' a,,{4 + NSNRi, + NSNRi, - 2N^SNRi,SNRi, cosa.J 



o'„^. 



where: />„=! + NSNRi, + NSNRi, + N' SNRi,NSNRi, sin' a,^ (2.20) 

Expressions for the numerator of equation (1.10), the output SNR, are possible for the 



two users: 



I |2 /?' r 

r>o\ =-ZTJ^[^yfsmI{l + NSNRi,,sm' a,,)+N,fsm;;cosaJ (2.21) 

Substituting equations (2.20) and (2.21) into equation (1.10) wiU result in the steady-state 
output SNR for the two users at the point of minimum mean-squared error. 

An additional substitution might simplify the expressions further. If we substitute the 
expression SNRij= a^ SNRi2 into equations (2.20) and (2.21) the output SNR can be 
expressed as a quotient of polynomials in a with coefficients that are functions of NSNRi2 
and ai2. The numerators of the quotients are: 



num, = N'SNRi^ll + NSNRi^ sin' a,^f 



cosa,- 



n2 



l + NSNRi^sm^a^2j 



num. = N'^SNRi^, sin^ a 



2 , cosa,2 



(2.22) 
n2 



NSNRi^ sin ' a^^ NSNRi^ sin ' a .^ 
for user 1 and user 2 respectively. The denominator of both biquadratic temis 



33 



Denss], = a^N^SNRi^ sin' a,,[4 + NSNRL,{a^ - laco&a,, + 1)] + 

NSNRi^la'' + 2acosa,j + 1) 

and the expressions for the two output SNRs become: 



(2.23) 



SNRo, = a'SNRi, ^^^^ SNRo, = SNRL -^^^^^^ (2.24) 

^^^sm Denssu 

The expressions for output SNRs may be interpreted as transfer functions in the 

variable a. The numerators may be treated as products of second order polynomials with 

easily determined roots which may be interpreted as real zeros of the output SNR "transfer 

function." The roots of the numerators are: 

cos a, 



z =0 ^12 



1 + NSNRL sin' a 



12 



(2.25) 



_ cosa, 

"Bum 2 



>sci:,, r I -. 1 

oA/CA/icT^ V±^^^-^NSNRi,xm'aJ 

INSNRi^sm a^^y 2 i2j 

The denominator is a fourth-order polynomial in a and analytical expressions for its roots 
are not particularly useful in the general case. To circumvent this difficulty the analytical 
expressions for output SNR will be examined and approximated for specific cases of a^ 
(the near-far ratio) and 0^2 (the angle between the phase-shift vectors in weight-space) 
and results will be compared to numerical solutions. Note that ai2 is determined 
exclusively by the users' DOAs for a given antenna array configuration. The output SNR 
expressions will be examined for four scenarios of the variable a: 

^•^ =0 (single user system) 

2. < a2 « 1 (1 vitdk, 1 strong user) 

■^- ^ = 1 (2 strong, equal power users) 

4. 1 < a2 < 00 (2 strong, unequal powers) 



34 

Two cases of DOA separation will be examined for each of the four scenarios listed 
immediately above. One case (referred to as case a) will be that of spatially orthogonal 
users (cos2ai2 = 0, sin2ai2 = 1). This will allow comparisons witii the tiie results in the 
previous section. The second case {case b) will be for DOAs with spacings greater than 
the Rayleigh limit (0 < cos2ai2 « sin2ai2 < 1) and will also be referred to as well- 
separated. 

The following analysis will consider approximations to equations (2.22)-(2.25) for 
some combinations of the intervals of a listed above. Ratiier tiian being exhaustive tiie 
analysis will focus on critical points which will reveal trends in performance. 
Single-User System 
When a = we have a single user system and : 



SNRo, = a^SNRi 



Den 



= 



SNR 



(2.26) 



SNRo, = SNRL -^^^^^^ = 



Den 



NSNRL 



SNR 



One Weak User. One Strong U.ser 

For the case of one weak user and a moderately sti-ong user which are well separated 
(case b) die numerators and denominator of equation (2.24) can be approximated by: 



num, = N^SNR^l + NSNRi^ sin' a,J 



a + - 



cos a, 



12 



l + NSNRi^sin^a^^ 



(2.27) 



num^ '^N'^SNRi^ sin'' a^^ 



li 



a' + 



NSNRi^ sin' a,2 



35 



Derisffn = N^SNRil(4+ NSNRi2)sm^ a,^ 



a' + 



NSNRi^ (4 + NSNRi^) sin^ a,^ 



If we can further assume that NSNRi2 » 4 then the output SNRs may be approximated 



by: 



SNRo, =a^N- SNRL sin^ a 



[a + (cosa„)z^,f 



[^'+^2^]' 



SNRo^ = N • SNRi^ sin' a,2 7 t- 



(2.28) 



where: ^.^ = 



P2b = 



_ ^2* 



NSNRi,sm'a,, ' *''" N' ■ SNRi', sin' a,, NSNRi, 



Where the notation Zjb represents the numerator zero for scenario 2 and DOA case b. 
Note that pzb < Zjb- Also note that for scenario 2, SNRoj is smaU since it is multipUed by a 
second order zero at a = 0. As a increases, SNRoj increases with some possible wrinkles 
in the magnitude curve added by the numerator zeros ( a = Z2bCosai2) and denominator 
poles ( a = yfp^). Note that since the cosai2 term in the numerator of equation (2.28) 
can be less than zero the possibUity exists that SNRoj could equal zero for nonzero a. The 
most interesting behavior is displayed by SNR02. For very small a, SNR02 is near the 
maximum value SNRo2= NSNRi2. As a increases away from zero, the SNR02 function rolls 
off at the -3 dB point of a^ =p2b and continues to decrease until a^ = Z2b ( SNRii = 
[Nsin^ai2\-^ )• At a^ = Z2b the SNR02 roll-off attenuation is halted and SNR02 begins 
increasing since die pole z.ip2b is canceled by the zero at a^ = Z2b. Over the range of a 
specified by this scenario, the point a^ = Z2b represents a minimum of SNR02. Substituting 
a2 = Z2b into equation (2.28) gives SNRoj = 1 and SNRo^ = 4NSNRiJ{NSNRi^ + 5) - 4 . 



36 

Note that similar expressions for the case of orthogonal users ( case a above) can be 
readily found by simply setting sin2ai2 = 1 and cos2ai2 = into equation (2.28). 

In summary, for most of scenario 2 the array is unable to resolve the low-power user 1 
input signal from the input noise. The array can just begin to resolve the user 1 input signal 
from die input noise when a^ = Z2b ( SNRii = [//sin2ai2]-' )• At this point, SNRoi = 1 and 
SNR02 = 4 which represents a minimum value of SNR02 over this scenario. 
Two Equal-Power Users 

If the two users have equal input power levels die output SNRs are equal: 



f^^f^Ri (1 + cosa,2 + NSNRi sin' a,, Y 
SNRo = ^^ — 

2 1 + cosai2 + NSNRi sin' a,^ \2 + NSNRi{\ - cosa,^ )) 



(2.30) 



and for die case of well-separated users (case b: < cos2ai2 « sin2ai2 < 1) die output 
SNRs may be approximated as: 

NSNRi (NSNRi sin^ a,^f 



SNRo == 

(2.31) 



2 NSNRi sin' a,^ (NSNRi{l - cosa^^ )) 



NSNRi I X 

= -^— (l-hcosajj 

If die users are spatially orthogonal {case a) dien sin2ai2 = 1 and cos2ai2 = and die 

output SNRs are exacdy equal to SNRo = NSNRi/2, which is consistent widi equation 

(2.6). 

Two Strong ILsers 

For diis scenario (#4) we have two strong users widi unequal input power levels. 

For case b ( < cos2ai2 « sin2ai2 < 1 ) die numerator terms of equation (2.26) 
may be modified by multiplying all SNRi2 terms by an additional sin2ai2 tenn. The 



37 



denominator (equation (2.23))changes slightly: 



A, =aVa^ 



+ 1 



N' SNRi; sin' a ^-, NSNRL, J N^SNRi^&in^a^^ 



(2.32) 



Terms containing higher powers of a dominate the numerators (equation (2.22)) and 
denominator (equation (2.32)) of the output SNR expression (2.24). This condition 
represents the presence of two sti-ong users of unequal power: SNRoj » SNR02. The 
expressions for the (scenario 4, case b) output SNRs may be approximated as: 



SNRo. = ^^ ^— — i^ 

NSNRi^sin^a^^ 

SNRo, --NSNRLsm^a 



(2.33) 



12 



Note tiiat for large input SNR the expressions in equation (2.33) should be nearly equal. 
The difference between die output SNRs for tiiis scenario is: 



ASNRo = SNRo, - SNRo^ = 2 + 



NSNRi^sin^a^^ 



= 2 + - 



1 



SNRo. 



(2.34) 



Which is very similar to the expression for output spread for orthogonal users given by 
equation (2.14). 



Numerical Results 

This section will first present numerical solutions to corroborate the two-user analysis 
of die previous section. The data will be presented in plots of the output SNRs (equation 
(2.26)) versus Uie input near-far ratio (a^). Array performance will be evaluated for 
spatially orthogonal users and well separated users evaluated over a wide range of a^. 



38 : " , 

Specific comparisons between analytical and numerical solutions will be made for critical 
points identified in the analysis. 

Performance for a linear MMSE array was investigated by plotting the output SNRs 
of two signals while varying one signal input level and holding the other at a fixed value of 
input SNR. For each point the MMSE weights and corresponding output SNRs were 
calculated using equations (1.6)-(1.10) and equation (2.1). The DOAs were fixed to 
specified values. The plot below shows the output SNRs for the case of a three element 
array with two users. User 1 is positioned broadside to the array (DOA = degrees) with 
varying input SNR, and user 2 is positioned at 41.8 degrees from broadside with an input 
SNR of 10. This DOA condition gives the least amount of source separation which allows 
the sources to be spatially orthogonal. 

Figure 2.2 shows the (A^ = 3) array output SNRs versus a' for a two-signal scenario 
over the range 10-5 < a^ <\^. The users are spatially orthogonal, ai2 = 90 degrees and 
the user 2 input SNR is 10. For a^ = 10-5 the array is in essence a single user system. 
SNR02 = mNRi2 = 30 which is die maximum possible output SNR for user 2 and SNRoi 
= 0. As a {SNRii) increases, SNR02 begins its roll-off attenuation: SNR02 rolls off 3 dB by 
the time a^ = .001 1, as predicted by equation (2.28). As a increases the SNR02 roU-off 
continues until it reaches the minimum value of 3.33 at a^ = 0.033. At this point the array 
is just able to resolve signal 1 from the input noise and SNRoi = 0.89. These numbers 
agree well with equations (2.10) and (2.28). 



39 




Figure 2.2 Output SNRs versus near-far ratio for spatially orthgonal incident 
signals 

For the scenario a^ = i the output SNRs are equal to 15 (see eqn. (2.6)). As a^ 
becomes very large the output SNRs lose their dependence on a^ and approach nearly the 
same value. This is shown in equations (2.33) and (2.34). At large near-far ratios (a^ = 
10 ) SNRoi = 32.03 and SNR02 = 30. The difference between their values is ASNRo = 2.03 
which agrees precisely with equations (2.14) and (2.34). 

Figure 2.3 shows the performance for DOA spacings greater than the Rayleigh limit 
For this scenario user 1 remains at array broadside and the DOA for user 2 moves to 90 
degrees (in line with the array). This case gives the vector quantities VjV2= Ncosai2= - 
\, cos^ai2 = 1/9 and sin2aj2 = 8/9. When a^ is smaU (a^ = 10"') SNR02 = NSNRi2 = 30 
as in the previous case. As a^ increases to 1.18x10"^ SNR02 rolls off approximately -3 dB 



40 

to a value of 15 and SNRo] increases from nearly zero to 6.5x10 ^ Note that equation 
(2.28) approximates SNRoj = 5x10'' and a -3 dB roll-off point of a^ = 1.25x10 ^ 

At a2 = 0.042, SNRo, = 0.798 and SNR02 takes on a minimum value of 2.807. 
Equation (2.28) predicts a minimum value of SNRo2= 3.87 will occur for a^ = zib = 
0.0375 and SNRo, = 1.01. From the plot SNRo, = SNR02 = 9.94 when a^ = 1 which 
agrees with the approximations given by equation (2.31). When a^ increases beyond unity 
the outputs quickly level to values near the maximum of SNR02: SNRo, = 28.64 and 
SNR02 = 26.61 at a^ = 1x10*. This gives an output spread (eqn. (2.34)) of ASNRo = 2.03. 

Figure 2.4 shows what happens when the DOA spacings are less than the Rayleigh 
limit In this case, signal source 1 remains at broadside while source 2 moves to DOA2 = 




Figure 2.3 Output SNRs versus near-far ratio for nonorthogonal incident signals 
with DOA spacings greater than the Rayleigh limit 



41 



20 deg., about half the Rayleigh limit. The outputs are degraded for a^ » 1, compared to 
the two previous plots. 



Output 
SNR 




10^ Near-Far Ratio: a^ 10* 



Figure 2.4 Output SNR versus near-far ratio for incident nonorthogonal signals 
with DOA spacings less than the Rayleigh limit 



CHAPTERS 

THE MULTIUSER ARRAY PROCESSOR 

ADAPTIVE PERFORMANCE 

Consider for a moment that filter weights in an adaptive process are random variables 
with salient statistical properties. Adaptive processors are unable to perfectly track the 
corresponding steady-state solution; this induces a penalty known as misadjustment'^ The 
goal of this chapter is determine the effect of the step size of the complex LMS 
algorithm on the steady-state SNR-leveling performance of the multi-user processor. 
This concern is motivated by two factors. First, from an intuitive standpoint, it seems that 
the multi-user processor might cease to level the output SNRs if the misadjustment 
becomes too large. Second, the multi-user array processor might share some similarities to 
an LMS automatic line enhancer (ALE) for multiple time signals. Fisher and Bershad" 
studied the misadjustment performance of an LMS-ALE for the case of multiple sinusoids 
in AWGN and found the equaUzer misadjustment to be especially sensitive to the step size. 

This work will rely heavily of the work of Senne"^ who developed expressions for the 
time-dependent and steady-state weight covariance matrix of a real LMS adaptive 
processor. Other authors have investigated the transient and steady-state behavior of the 
complex LMS weight covariance matrix.^^'^^ and have found that the steady-state 
eigenvalues are very similar to those derived by Senne for the real LMS algorithm. This 
detail will be applied in a later section of this chapter. 



42 



43 

Other authors have examined the performance of adaptive processors subjected to 
hard and soft constraints on SNR as well as optimizations of SNR itself subject to 
nonlinear constraints.'*'' 

This chapter will be divided into three sections. The first section will give the 
development of a performance measure (cost function) which is based on the mean output 
SNR of the adaptive array. The second section develops a performance measure based on 
the variance of the array output SNR. The last section presents numerical results which 
allow comparisons between the analysis and simulations. The cost functions will be used 
to find acceptable bounds on the LMS step size. It is found that the upper bound on step- 
size arising from stability arguments also gives acceptable output SNR performance for 
the multi-user processor. 

Mean-Based Performance Measure 

If the adaptive weights of an LMS processor are considered to be time- varying 
random variables, w(n) = Wo -h v(n;, the mth user's time-dependent output SNR results 
from substituting the expression for w(n) into equation (1.10): 

|(Wo+v(/i))'"u 



SNRo{n)„=SNRi„ 



(3.1) 

The random, time-varying weight component is represented by v(n), the mean of the 
weight vector is the steady-state MMSE weight vector, Wo = R"^p, and the discrete time- 
dependence is introduced by the variable n. Note that the phase-shift vector u„ and SNRi„ 
are constants. 



44 

In order to derive an expression for the performance measure we need to find the 
mean of the output SNR. Some assumptions facilitate this effort: 

1. ||v(n) + w„|| = ||w„|| ; instantaneous deviations of norm(w) from 

norm ( Wo) are negligible because of small step size ^i. 

2. v(/i) are independent from sample-to-sample. 

3. Input signals and noise are i.i.d., stationary and ergodic. 

4. The elements of w(n) form a jointly Gaussian process. 

The resulting mean of the output SNR is: 

E{SNRo{n)^ ] = SNRi„ ■ ^^ + SNRi„ ■ ";^-"" (3.2) 

W W 



where the first term on the right-hand side is SNRom , the steady-state output SNR defined 
earlier in equation (1.10). The matrix Cw = cov(v(n)) is the steady-state weight covariance 
matrix given by Senne.'' The matrix is not a function of time, n, since v(n) components are 
i.i.d. 

The last term on the right-hand side will cause the average output SNR for user m to 
increase as the adaptive weight covariance increases. This might have Utde effect for weak 
users; the term is not negligible for strong users. The average output SNRs may no longer 
be leveled if the term becomes too large. This point is clarified in Appendix A which 
presents some general characteristics of the SNR performance surface. 

The first measure of performance is the difference between the average and steady- 
state output SNRs divided by the steady-state output SNR: 

E{SNRo{n)„]-SNRo„ _ SNRi^ nlC^y„ 

SNRo„ SNRo„ ||w„f - • ^^-^^ 

Where SNRom is the steady-state output SNR for user m and is defined by the first term on 



45 

the right side of equation (3.2) above. The quantity Ci »= l/(NSNRi2) « 1 is a constant 
which serves as an upper bound which will limit the excursions of the time- dependent 
SNR (resulting from the adaptive process) from the steady-state solution. The constant 
will be defined in more detail later in the analysis. If the adaptive process is allowed to 
stray too far from the steady-state solution the output SNRs may no longer be held to 
nearly the same level and near/far limited performance may result. Using this relationship 
with equation (3.2) above gives an expression in which step size dependence is expressed 
indirectly through Cw: 

T^ • ^ /^ II l|2 S^^O„ 

u«C„u„ < C, wj (3 4) 

SNRi^ 

The quantity on the left-hand side may also be lower-bounded by using the maximin 
theoreom^^: 



mm 



mm 



'r(Aj = qeC- 
q?tO 



V 



(3.5) 



q q J 

where C^ denotes the complex A^-dimensional space, q is an A^-length vector and ?lw = 
minfaU eigenvalues of €„}. Let q of equation (3.5) equal u„ of equation (3.4) where 
Um Utn*=N. Equation (3.5) may then be used to form a lower bound for equation (3.4). 
The left-hand term of equation (3.4) would then be bounded by: 

'V-'^V)^"IC„u:.C,|kr^ (3.6) 

The next task is to find a useful expression for the lower bound of equation (3.6). 
Senne"^ has shown that the weight covariance matrix is diagonalized by the eigenvector 
matrix of R, the input autocorrelation matrix. For an ^ = 3 array, the diagonal of Av^, the 



46 



eigenvalue matrix of Cw, is: 



diag{A„} = 



^K 






1-1 

2 



( M, 



U-M 



ly 



M2 "I 
l-;iA 



2 J 



' M3 ^ 



U-M 



3/ 



(3.7) 



where A;, A2, and X3 are the eigenvalues of R, and |i is the LMS step size. Compton'"*' has 
shown that when SNRi] » SNRi2 > 1, the signal and noise eigenvalues of an N-by-N 
autocorrelation matrix R are approximated by: 



A, =(T^(l + A^-5A7?i,) 
f ( 



K-o] 



K=ol 



l+SNRL 



V 



N- 



I T '\^W 



N 



(3.8) 



yj 



m = 3...N 



where the largest eigenvalue A/ is established by the strongest incident signal and A™ are 
the noise-only eigenvalues. 

If the step size is small (ji < 7/A;)the smallest Aw from equation (3.7) may be 
approximated by: 



"'"{^4=-^(i-M,) 



1-- 



/iA, 



If ^X, 



1-//A, 



if nX, 



1-//AJ 2ll-/iA 



3y 



(3.9) 



;^e: 



The simplification m equation (3.8) results directly from our assumption of small step size. 
If jj, is small the term in brackets and the term in parentheses outside of the brackets are 
equivalent to unity. 



47 



This establishes the upper and lower bounds on u'^C u* 



m vv m 



^mel,^ < uXX < C,Kf ^^ (3.10) 

^ SNRi„ 

If all the incident signals are well-separated, strong and perfectly correlated to the 
reference waveforms the minimum MSE should be well-approximated by the output noise 



power: e^, = al ||wj . If user m = 1 is the strongest user then SNRii can be determined 
in terms of the eigenvalues via equation (3.8) (N- SMRi^ = (A,/<t^) - 1 = A,/ct^ ). The 
constant Ci from equation (3.3) establishes a bound on the average excursions from the 
steady-state output SNR due to misadjusttnent, so it must be made suitably small. The 
strongest user - user 1- should have a steady-state output SNR leveled to approximately 
the same value as user 2, therfore SNRoi = SNR02 = NSNRi2. Since even SNRii » 1 
then Cj = nJO-NSNRii) should give acceptable results where |io is a constant (0 < //<, < 1) 
and represents die unormalized step size . This results in an upper bound on //: 

which is similar to the upper bound on the LMS step size derived from stability 

22 93 J • 
arguments, • and is consistent with our earlier assumption of a small step size. 



Variance-Based Pe rformance Mea.snr^. 

The second cost function is the variance of the adaptive output SNR divided by the 
squared mean of the adaptive output SNR. Choosing \i to keep this measure small will 
limit excursions from the mean of the adaptive process. The second moment of output 
SNR is: 



48 



E[SNRoin)i] = ^-d{y,„+y{n)fuS] (3.12) 



Senne assumed that the filter weights, w(n) = Wo + v(n), form a jointly Gaussian process. 
In the following analysis this assumption is used to expand fourth-order joint moments in 
terms of the second-order moments via the Gaussian moment factoring theorem.'"' 

If we assume the vector v(n) is multivariate and complex Gaussian we can express the 
fourth order moments as functions of the second-order moments.*"' Expanding the 
bracketed vector term and discarding terms that would give odd-ordered moments of v(n) 
results in several terms: 

4|K + v)'u„p] = |wXf + 4 • |w>„|'<C„u: 

+ 4wX)»:)%(wX)»„f] (3.13) 

N N \ S 

where we have dropped the time-dependent notation from the weights. The third term on 
the right is zero. The fourth-order moment may be expanded in terms of the second-order 
moments: 

4^ K ^ K ] = 4^'. < ]4^ K ] + 4^ ^i ]4^ ^ ] (3. 14) 

The simimation term becomes: 

IIII"',<"/,"/:4^^^K]=2-Kc,,u:)^ (3.15) 

^-1 1,-1 1,-1 1^=1 

The second moment of the output SNR is: 



'^011 



(3.16) 



49 



and the variance of the output SNR is (from equation (3.2)): 

|wXf+u:C„u:)'-|wXf 



f^SNRoJin) - SNRi„ II ||4 



w 



oil 



>l 



= E[SNRoMY - SNRol 

where the second line of equation (3.17) follows from (3.2) and SNRo„ is the steady-state 
output SNR of the m'th signal. The performance measure becomes: 



^SNRo(n)_ , SNRoi 



= 1-3 TT-^- ^ C2 (3.18) 



45A^^o(n)J £[5A^^o(az)J 

where C2 « l/(NSNRi2) « 1 is a constant to be defined later. 

Substituting quantities from equation (3.2) into equation (3.18) results in the 
inequality: 

SNRo 



u^C u* < llw 



m ^ vv m 



VT^Q 



' -1 



(3.19) 



Since C2 < 1 the radical term may be approximated by a two-term series 
(1 - C2) 2 = 1 + C2/2 and the inequaUty takes the same form as equation (3.4). The same 
procedure used to determine the step size upper bound for the first performance measure 
may be followed for this second performance measure with the same results: /i < ^JXi . 

Simulation Result.s 

This section will use simulations to corroborate the steady-state analytical results of 
previous sections. 



so 

The next two plots show curves of the quantities obtained from equations (3.2) and 
(3.18). Figure 3.1 shows the average output SNRs versus the step size coefficient for 
means resulting from simulation and from expression (3.2). The input scenario is SNRii= 
10, SNRii = 10^ DOA] = deg., DOA2 = 90 deg., A^ = 3. The initial weight for the LMS 
simulation was MMSE weight vector from equation (1.8), Wo = R'^p. 

Plots of the variance-based performance measure are shown in Figure 3.2. The input 
scenario is identical to the one described in Figure 3.1. The curves show the ratio of 
variance divided by the squared mean of the output SNR obtained tiirough analysis 



40 



35 



Average output SNR vs. step-size coefficient 



m 



3 30 
o 



I 



25 



20 



I I I ip 



SNRi1 = 10 dB 
SNRi2 = 60 dB 
D0A1 = deg 
D0A2 = 90 deg 



User 2: sim 
I I Ill ll-H 




User 2: calc 

I M III III III II II II III I n il II II II II II lllll II III 

User 1: sim, calc 



10^ 



10-* 10-' 

Step-size coefficient 



10" 



Figure 3.1 Output SNR versus step-size coefficient |io. The step size is 
normalized by the input power 



(equation (3.18)) and simulation. The curves show, as we might expect, that the 
performance will be worst for the strongest user. Selecting step size for acceptable strong 
user performance will lower-bound the performance for the remaining weaker users. 



51 



10' 



2nd moment perfomiance measure vs. step size coeff cient 




10" 



lO" 



10" 



Step size coefficient 



10" 



Figure 3.2 Variance-based estimator versus step-size coefficient (Xo. 
The step size is normalized by the input power. The input scenario is 
identical to the one described in Figure 3.1 

Selecting a small performance measure will limit the point-by-point excursions of the 

output SNR from the steady-state value and will give results which are acceptable in the 

mean. If 1% is selected as an acceptable value of the performance measure then from the 

curves for user 2 above it would appear that ^ = 0. l/Tr(R) is the maximum allowable step 

size. This is in agreement with existing rules-of-thumb for selecting step size for stability 

and convergence. For these conditions, it would appear that upper bounds on LMS step 

size derived from convergence arguments will give acceptable performance of the 

multiuser processor. 



CHAPTER 4 

BASE STATION RECEIVER PERFORMANCE USING A SMART ANTENNA 

ARRAY IN A DS-CDMA SYSTEM WITH IMPERFECT POWER CONTROL 

This chapter examines the steady-state outage probability and outage-based capacity 
of a single cell containing multiple directional signal sources transmitting to a central base 
station. The only fluctuation in the signal levels is due to lognormally-distributed power 
control error in the multiple transmitters. The central receiver consists of an array of N 
isotropic sensors, K minimum mean-squared error single-user beamforming processors 
and a bank of conventional detectors. The performance measures are outage probability 
and outage-based capacity. The goal is to find simple expressions which relate the outage- 
based capacity to the antenna array parameters, the number of active signal sources and 
the degree of power control error. The analytical results are expressed in terms of the 
number of users per array element which may be supported for a given outage probability. 
Analytical results are found to agree closely with those obtained from Monte Carlo 
simulations. 

This work is unique in several respects. First, most previous work examined uplink 
performance for a traditional single-channel receiver. Second, the authors which 
considered a base station antenna and imperfect power control presented simulation 
results for the case of closed-loop power control.*^*^ As was mentioned in Chapter 1, 
power control error will be modeled by incident power levels which are lognormally 
distributed. This model has gained some acceptance for open-loop power-controlled 



52 



53 

systems. Its suitability as a model for closed-loop power-controlled systems remains an 
open issue. 

The remainder of the chapter is divided into several sections. A qualitative description 
of the array and the receiver is given in the next section. The third section outlines the 
development of the analytical model and the derivation of the expressions for uplink 
capacity. The fourth section describes the Monte Carlo simulations and compares the 
analytical and the simulated results. The last section presents the conclusions. 

Svstem Description 

The incident DS-CDMA signals are independent and originate from independent 
transmitters which are arbitrarily placed about the base station receiving antenna. The K 
active transmitters (located in a single cell) result in incident signals with independent 
directions-of-arrival (DOA) uniformly distributed over the interval [0, lit). The 
transmitters' output levels are continuously adjustable and have an infinite dynamic range. 
The output power - adjusted by an unspecified power control algorithm - results in output 
power which is independent between sources and which is also lognormally distributed. 
The modulation format is BPSK. 

The base station receiving array consists of A^ ideal isotropic sensors arranged in a 
circle. Adjacent sensors are separated by a distance with the electrical equivalent of one- 
half wavelength. The sensor array feeds K separate banks of ^V complex weights 
controlled by K MMSE beamforming processors. The optimum steady-state weight 
vector for each beamformer is given by equation (1.8), Wo = R'^p. Each of the K 
beamformer outputs feeds a distinct conventional DS-SS detector which in turn provides 



54 

an estimate of the demodulated output. The receiver is assumed to be capable of perfect 
carrier tracking of the desired signal. Unlike the multiple-user array of the previous 
chapters, the array/detector combination is devoted to detection of a single desired signal 
amidst multi-access interference. A diagram is shown in Figure 4.1. 




Conventional 
DetectOT 1 



bat,] 



Conventional 
DetectOT 2 



Conventional 
DetectOT K 



I'ea.K 



Figure 4.1 A bank of baseband single-user beamformers and conventional detectors 
sharing an array of sensors 



Analysis 



The stationary, complex base-band output of the sensor array is found by combining 
equations (1.3) and (1.4): 



55 



m = At,iO''"'c„it-r„)b„{t-T„)u^+n{t) (4.1) 



msl 



where A is the peak amplitude and €„ is N(0,apc/). The sensor outputs are weighted and 
summed by the beamformer and then processed by the conventional detector which forms 
an estimate of the current output bit. The signal from source 1 is considered the desired 
signal. The remaining K-1 signals are considered multi-access interference. 

The array tends to steer a pattern lobe towards the desired signal and also tends to 
steer nulls of finite depth towards the N-2 strongest interferers when K >N-L Because 
the strongest signals are attenuated, no single signal dominates the output statistics and the 
output may be well-approximated as Gaussian. An additional assumption of long codes 
(codes which span more than one data symbol) allows the PN sequences to be modeled as 
random codes. The random code model of Pursley'°^ will be used here. More refined 
models exist, but the additional complexity they introduce tends to improve accuracy 
when only a few users are present. '°3-'**-io5 

From Pickholtz et al.'°* we know diat the effective bit energy to noise spectral density 
ratio may be approximated by (Eb/No)eff^NcSINR. The random code approximation may 
be modified to account for varying input amplitudes among the incident signals due to 
their power control error. Using one of the intermediate steps (equation (14) from 
Pursley' ) allows a convenient expression for {Eb/No)eff while retaining the cross- 
correlative properties of the PN codes: 



^ E ^ 



X, c-rxrr, N.SNRo, 

= N^SINR = — — j^ ! (4.2) 



c m=2 



where SNRo„ is the array output SNR of the mth incident signal and r;,^; is a 



36 



crosscorrelation parameter between the first and mth PN codes. If SNRom is replaced by its 
sample mean and the remaining sum of r„^i terms is replaced by the random code 
approximation (equation (16) from Pursley'"^) the expression becomes: 



El 

N. 



NSNRo, 
= Y^ ■ (4.3) 

^ m=2 

The quantity SNRok may be rewritten as: 

SNRo„=\Q''"'.Gp„SNRi (4.4) 

where Gp„ = |wju„| /||wj| is the normalized power gain of the array towards the mth 
signal and SNRi = A^/On^. 

The power gain is a complicated function of the input scenario and the array 
geometry. In order to simplify the analysis we will use a simplifying assumption that the 
array power gain with respect to the first user, Gp, = Gpa, wiU be approximately equal to 
its upper bound A^, the number of array elements. The MMSE array gain towards the 
interferers (Gp„,m = 2 ... A) is difficult to characterize. Intuitively, we might expect that 
the avg{ Gp„ } = Gpi might be well-approximated by the average (over DO A) normalized 
power gain (Gpayg) of a classical beamformer (w, = uj with no adaptive null-steering 
capability. The value of Gpa^g ranges from unity for a single element to 1.6 for a 30- 
element circular airay. Simulated results presented shortly will show that these 
assumptions with regard to the individual quantities (Gp^, Gpd are not always valid over 
the conditions of interest. The average of die ratio GpJGpi .however, does provide a 
reasonably good fit to die simulated results when die above assumptions (Gpd - M Gpi « 
N) are used. 



57 



If there are many interfering signals the sum in the denominator of equation (4.3) may 
be approximated by averages: 



r T7 \ 



E, 



N„ 



NMSNRilO 



x/\0 



^w-, (4.5) 

V"" y^ ^K - l)SNRiGpilO ^oo ""' + 1 



Further explanation is required. The sum in the denominator of equation (4.3) is almost a 
sample mean and may be approximated by an ensemble average. Assume the power gain 
and the exponential of equation (4.4) are independent: the expectations apply separately. 
This results in the base- 10 exponential term, the (K-1) term and Gpi in the denominator of 
(4.5). This approximation causes a slight overestimation of interference: by breaking a sum 
of products into a product of sums we have invoked Schwartz's inequality twice in 
succession to get from (4.2) to (4.3) and to get from (4.3) to (4.5). Our simplifications 
have equated sample means and ensemble averages and also ignored the complicated 
nature of the array response by treating the power gain as an averaged quantity. 

Simulations of the power gain show our assumptions regarding Gp may give 
acceptable results. Figure 4.2 shows the results of Monte Carlo simulations of averaged 
values of (MMSE) Gp for a desired signal (Gp^), an interferer (Gp, ) and their quotient for 
a varying number of users, 3 dB of power control error and a 30-element array. The 
curves show that as the number of users exceeds the degrees-of-freedora of the array our 
simulated values for the individual gains are not very close to approximated values of Gpd 
= iV = 30 and G/?„ = Gpavg = 1.6 but their quotient GpJGpi, = A^ = 30. So, for this 
circumstance it might be useful to approximate the gains in equation (4.5) as Gpd = A^ and 
Gpi, = 1. 



58 

Why will this work? As the number of users increases and the system becomes 
interference-limited the "1" term in the denominator of equation (4.5) becomes negligible 
leaving a good approximation of the quotient of the gains as Gpd/Gp, ~ N. When there are 
few users the multi-access interference is negligible, Gpd ~ N and the approximation will 
still hold. These gain approximations do not hold separately to predict the desired signal 
output power or the interference output power, but their combination might prove useful 
in calculating outage. 

What about more extreme scenarios? Figure 4.3 shows the gains and the gain ratio for 
a power control error of 10 dB. The individual approximations Gpd « A^ and Gp„ = Gpavg 
= 1.6 are even worse than before. However, the simulated gain ratio Gpd/Gpi = N/L3 for 
K>N. This is a little closer to the ratio of the individual gains Gpd /Gpavg . = N/1.6. This, 



1Cf 



10" 



Gain 



ICP 



ia' 



1 1 1 


' 


^/^ — ^— — ^ 


Gp/Gpi 


^""^^^ 




■ 


Op4 


■ 


Gpi 


power control error = 3 dB 

1 1 1 


N = 30 

L. 1 



20 40 60 80 

Number of Incident Signals 



100 



120 



Figure 4.2 Gain versus the number of incident signals. Power conu-ol error is 
3dB. 



59 



^(f 



QP..t:.K=M. 




Gain 



^(f 



^Q■' 



power control error = 10 dB 



N = 30 



20 40 60 80 

Number of Active Users 



100 



120 



Figure 4.3 Gain versus the number of incident signals. Power control error is 
10 dB. 



and other simulations, indicate that choosing Gpd = A^ and Gp, = Gp^v^ = 1.3 for this 
model is a good approximation for outage calculations over the ranges of power control 
error examined here (0 < Opec < 10). Note that Gp, = Gpa., = 1.3 also arises from a 
sample mean (over A9 of the average power gain (over DOA) for A^= 1,2,4,8,15 and 30- 
element arrays for a classical beamformer. 

Approximating the interference as an average quantity in equation (4.5) eUminates 
complex scenario-by-scenario interactions between the desired signal and the interference 
in the analytical model. It also leaves the lognormally distributed desired signal as the only 
random variable in the model. The resulting distribution of the (Eb/No)eff in equation (4.5) 
is therefore log-normal and results in simple expressions for the outage probability and the 



60 



capacity. Outage occurs when the {Eb/No)es is less than some threshold. Converting 
equation (4.5) to dB and noting that ei is N(P,<J^pcc) results in a simple expression for the 
outage probability: 



PTou,,,. =Pr 



W^" ^eff.dB 



< ^ 



dB 



= Q 



ZdB-^ 



dB 



(4.6) 



V /« / 



where Q(x) is the complementary Gaussian CDF, ^b is the desired threshold in dB and: 



Z,fl = 10*log 



10 







k 


D 

J 



(4.7) 



The quantity D is the denominator of equation (4.5). The quantity Eb/No « (Tb/Tc)SNRi 
=NcSNRi represents the equivalent bit energy to noise density ratio for a single incident 
signal and a single array element. Note that Naguib et al.^* has also developed a Q- 
function upper bound for outage probability for the case of perfect power control. For the 
case of imperfect closed-loop power control Naguib et al.*' presented the simulated means 
and variances of (Eb/No)eff. 

Some simple algebraic manipulations result in the average capacity as a function of 
desired outage probability: 



K = C 



where: 



f'« 



i — ^ 



(pf~».) 



v-l 



N. 



oj 



+1 



(4.8) 



IGp, 



where for high Et/No the capacity is approximately linear in processing gain, Nc, or the 



61 

number of array elements N but decays exponentially as the power control error increases. 
Note also that the threshold ^ is no longer in dB. Note that if we interpret equation (4.8) 
as being linear in N, we can define Si per-element capacity by noting the slope of the line. 
Note also that the capacity asymptotically approaches a finite maximum as Eh/No 
increases. A similar effect was noted by Naguib and Paulraj*^ for the case of a 2-D RAKE 
combiner at the base station receiver and perfect power control in the mobiles. They 
examined the capacity as the equal incident signal levels went to infinity and named the 
parameter asymptotic capacity. A simple expression for uplink capacity was also 
formulated by Suard et. al.'^ for a post-detection combiner. The model for power control 
error was restricted to a single user with an incident power level 10 dB higher than the 
other users. 

Simulations and Results 

Monte-Carlo simulations were used to corroborate the analytical results given by 
equation (4.6). Autocorrelation matrices for the desired signal, interferers and noise were 
generated by ensemble averages as in equation (1.6). Signal DOAs were uniformly 
distributed over [0, 2ic). The processing gain was 127 and Eb/No = 7 for a single antenna 
element and no power control error. For a single trial the MMSE weights were calculated 
via equation (1.8) and the desired signal, interference and noise power out of the array 
were then used to calculate (Eb/No)eff via equation (4.3). An outage condition was judged 
to exist for that trial if (Eb/No)eff <7dB (i.e. from equation (4.6), ^b=7 dB). The 
quantities were averaged over 20,000 trials for each combination of power control error, 
number of users and number of array elements. 



62 

Curves of the outage probability - ?x(SEb/No)eff < 7 dB) - versus the number of users 
are shown in Figure 4.4. The figure contains curves from equation (4.6) as well as the 
Monte Carlo simulations. Power control error is 4 dB. Note that for 20,000 trials, curves 
in Fig. 4.4 might be inaccurate for outage less than 10"^. 

Figure 4.5 shows the total array/receiver capacity versus the number of array elements 
with outage probability as a parameter. The curves are formed by plotting constant 
contours of a three-dimensional surface formed by ?T(,(Eb/No)eff < 7 dB) as it varies over 
K and A^. The solid curves show simulated results; dashed lines show the constant- value 
contours of equation (4.6) for power control error equal to 4 dB. 

The curves of Figure 4.5 show the user capacity of the array/detector is roughly linear 
in N. We may therefore use as a performance measure per-element capacity, the number 
of users per array element which may be supported for the given values of outage 
probability and power control error. As noted earlier the analytical expression for the per- 
element capacity may be obtained by noting the slope of the total capacity line in equation 
(4.8) with A^ as the independent variable. A point of note: the average power gain {Gpavg) 
in equation (4.8) has a slight dependence on A*^: it is equal to unity for a single element and 
is equal to 1.6 for a 30-element array. For the sake of simplicity, this slight dependence is 
ignored and a mean value of Gpavg = 1.3 is assumed, which was noted earlier when 
comparing simulated Gpd/Gpi curves in figures 4.2 and 4.3. 

The per-element capacity versus the power control error for Pri(Eb/No)^ < 7 dB) = 
0.02 is shown in the upper plot of Figure 4.6. For simulated data, the normalized capacity 
was determined by extracting the approximate slopes of die capacity curves (as shown in 
Figure 4.4) via a linear least-squares curve fit. The analytical results were calculated from 



63 



Outage 
Prob. 



N=15 



N=30 




Simulation 
Analysis, eq. (8) 



20 40 60 80 100 120 140 

Number of Incident Signals 



Figure 4.4 Outage probability versus the number of incident signals. Power 
control error is 4 dB. The number of array elements N is a parameter 



80 
Capacity 
70 




10 15 20 

Number of Array Elements 



Figure 4.5 Capacity versus the number of array elements with outage probability 
as a parameter. The power control error is 3 dB. 



64 



Per 30 

Element 

Capacity 

20 




10- 




Intercept 

-10h 
-20 
-30 
-40 



8 10 

Power Control Errw (dB) 



1 1 1 

• ■ ■«« 

........... I .......... J .... -^n.. . 


— '"/t 


, , - - -J.—-^- - • - 

^ — -r 


/ 


__ L„^-^' ; 


■ 


^Vy; i 


• 
• 




Simulations 


- Analysis, eq. (4.8) 



(b) 



8 10 

Power Control Error (dB) 



Figure 4.6 Normalized array capacity versus power control error in dB. 



the slope of the system capacity line given in equation (4.8). As the curves show, the 
analytical and simulated results for the per-element capacity are in close agreement. The 
lower plot of Figure 4.6 shows the intercept point of the line in equation (4.8). Obviously 
the analytical and simulated results are not in as close agreement as the upper plot. Note 
that to determine the overall system capacity - given an array of iV elements - it would be 
necessary to know the per-element capacity as well as the intercept 



65 

Conclusions 

In this chapter we attempted to develop accurate, simple analytical expressions for the 
outage-based uplink capacity in an idealized single-cell DS-CDMA system with multiple, 
possibly near/far, signals incident on a base-station receiving array. Power control error 
was assumed to be lognormally distributed. Some simplifying assumptions regarding the 
interference and the array response allowed an approximate expression for the uplink 
capacity that was compared with results from Monte Carlo simulations. 

The approximate expression - given in equation (4.8) - shows that a roughly linear 
relationship exists between the capacity and the number of array elements. The overall 
capacity K consists of two components. The first component is the slope of the line in N 
and has been defined as the per-element capacity, the number of users per array element 
which may be supported for a given level of outage probability and power control error. 

Equation (4.8) indicates that the per-element capacity is not dependent on the nominal 
input level (i.e. the input level without power control error: SNRi) but decreases 
exponentially with increasing power control error. For the levels of power control error 
examined here the term which dominates the exponential roll-off is (JpceQ''(Prou,age)/W of 
equation (4.8). The decrease in the per-element capacity is therefore dependent on the 
outage requirements and the standard deviation - in dB - of the power control error. Since 
the per-element capacity is insensitive to the nominal input levels and the array size, but is 
keenly dependent on the degree of power control error and the outage, it might serve as a 
useful asymptotic measure of the performance for this array/receiver. 



66 

The second component of equation (4.8) is the intercept term equal to l-C(Eb/No)'' 
where C is defined in (4.8). The intercept is inversely proportional to the nominal input 
levels. As the input levels decrease this term becomes larger, diminishing the overall 
capacity. In a plot of capacity (K) versus array elements (N), the capacity line moves away 
from the origin along the horizontal N^-axis as the nominal input levels decrease. The 
intercept is weakly dependent on the power control error (via Q and also on the number 
of array elements (via Gpavg) and is not dependent on the outage probability. Analytical 
and simulated results do not agree as closely as those for per-element capacity. 

The agreement - or lack of it - between the analytical and simulated results for per- 
element capacity and the intercept can be interpreted in terms of outage probability curves 
shown in Figure (4.4). The analytical model can predict well the horizontal spacing 
between the continuous outage probability curves. The per element capacity predicts the 
incremental increase in capacity with increasing array elements and is a measure of the 
horizontal displacements of the outage curves relative to one-another. The less reliable 
prediction made by the model is the horizontal placement of the outage curves relative to a 
point on the horizontal axis. This enters into the model via the intercept parameter 
described above. 

The simple model presented here was based on some simphfying assumptions 
regarding the array response towards the incident signals. In spite of this, the analytical 
model accurately predicts some aspects of the array/receiver performance when directional 
signals are employed with lognormally distributed power conttol error. The directional 
signals originated in a single cell and were not subjected to any kind of environmentally- 



67 

induced fading. The system performance in the presence of fading and interference 
resulting from outer cells is the topic of the next chapter. 



CHAPTERS 

BASE STATION RECEIVER PERFORMANCE WITH A SMART ANTENNA 

ARRAY IN THE PRESENCE OF MULTIPATH FADING AND SHADOW FADING 

This chapter examines base station receiver performance when incident signals are 
subjected to fnequency-nonselective Rayleigh multipath fading and lognormally distributed 
shadow fading. The effects of multi-access interference from outer cells - subjected to 
shadow fading - wUl also be included in the incident signal model. The power control 
error model will continue to be described by a lognormally-disttibuted random variable 
and the receiver still consists of an array of A^ ideal isotropic sensors, K minimum mean- 
squared error single-user array processors and a bank of conventional detectors (see 
Figure 4.1). 

The goal of this chapter is to determine the performance dependence on the number 
of active signal sources, the number of array elements and the degree of power control 
error. As before, the performance will be expressed by outage probability and per-element 
capacity. Unfortunately the introduction of fading and outer cell interference further 
complicates the development of simple analytical models. In spite of these complications 
the simulated results closely follow some general trends estabUshed by the model in 
Chapter 4. In particular, curves of outage-based capacity continue to be linear in A^ with 
slopes that decrease exponentially with increasing power control error. 

Previous authors have investigated the performance of optimum combining from the 
standpoint of interference rejection. Winters"'" and other authors^'^' have examined 
contributions to TDMA system performance while Naguib et al.^^** have studied CDMA 

68 



69 

systems. Unlike previous work, this research examines the outage-based capacity for the 
case of imperfect power control and attempts to provide some analytical models which 
would allow easy assessment of performance. 

The remainder of this chapter is divided into six sections. The first section will present 
the receiver model for single-cell signals subjected to Rayleigh fading. The second section 
will extend the model to the multi-cell case. The third section will quickly revisit the model 
of chapter four and introduce a second analytical technique. The fourth section will give a 
brief description of the simulation parameters and compare the simulated results with the 
analytical results.. The fifth section entitled Conclusions and Discussion will review the 
results in some detail. The last section gives a summary. 

A Single Cell with Signals Subjected to Ravleigh Fading 

Multipath fading arises when propagating electromagnetic waves originating from a 
single signal source arrive at a receiver via different propagation paths. The individual 
paths may include Une-of-sight propagation as well as paths resulting from reflection off of 
one or more surfaces. The individual waves - as well as their sum - are highly dependent 
upon the frequency of the propagating waves, their path lengths as well as the reflective 
properties and geometric arrangement of the encountered surfaces. 

This research will exploit the assumption that multipath fading of a signal from a 
single source results from the sum of many reflected waves. The individual waves have 
roughly equal power as well as independent amplitudes and independent phases. This 
allows the fading component of the incident signal to be modeled as a complex Gaussian 
random variable with uniformly distributed phase and an envelope which is Rayleigh 



70 

distributed.'"' If the channel is well-approximated by a constant frequency response 
characteristic then relative time delays between arriving wave fronts are negligible. This is 
known asfrequency-nonselective Rayleigh fading or flat Ray leigh fading. The flat-fading 
condition is probably an accurate approximation for indoor communication systems with 
large path losses and mobile systems with scatterers located in close proximity to the 
mobile. It represents a worst-case condition from the standpoint of detection since it will 
not allow the use of a RAKE receiver*^ to provide resolution of individual time-delayed 
paths. 

The complex base-band output of a sensor array outwardly resembles the expression 
given in equation (4.1): 

y(/) = AilO'--.c„(.-Tjfc„(r-T„K-Hn(r) (5.1) 

where the elements of the phase shift vector Um are complex Gaussian random variables 
which result from Rayleigh fading. This is in contrast to equations (1.4) and (4.1) in which 
the components of Um are complex exponentials resulting from directional, unfaded 
signals. 

The components of Um may have any permissible degree of correlation. The study of 
spatial diversity combiners is dedicated to antenna array structures which force the 
correlation of fading components between array elements to be low (ideally zero).'"* A 
base station diversity array must have larger element spacings than the customary half- 
wavelength spacings used in a mobile radio receiver.^* Lee conducted an empirical study 
of fading correlations in a two-element base station array. He concluded that, for low 
correlation, the interelement spacings must be 15X, - 20A, if the signal arrives from 



71 

broadside and 70A. - 80X, if the signal arrives along the axial direction. Salz and Winters'** 
examined a linear array and developed closed-form expressions for the direction- 
dependent fading correlations between array elements when multipath rays are "dense" 
throughout a range of DOAs. Raleigh et al."° proposed an analytical model which 
describes the spatially-dependent correlations of the fading process. Verification of the 
latter two models through experiment remains an open issue, as does more refined 
spatially-dependent channel models. Naguib and Paulraj*^ examined the effects of the Salz 
and Winters fading model on a base-station diversity array in an IS-95 system using 
closed-loop power control. 

Because spatial channel models remain an open issue this research will exploit the 
assumption that the fading process is independent between antenna elements and the 
elements of Um in equation (5.1) are complex i.i.d. N(0,1). Spatial dependence of the 
incident signals - via the interelement phase-shifts - no longer exists and the array 
geometry is critical only in that it results in independent fading between elements. 
Wmters has shown that even when directional information is not used by the processor 
(Le. the array functions as a diversity combiner) an optimum combiner will outperform, in 
steady state, a maximal ratio combiner because it is able to adaptively attenuate cochannel 
interference. 

Multiple Cells with Signals Subjected to Rayleigh Fading and Shadow Fading 

The models and results from the last subsection will be extended here to include the 
effects of outer-cell interference. Like the previous section of this chapter the incident 
signals will be subjected to flat Rayleigh fading. Unlike the previous section, however, the 



72 

cell which contains the desired user wUl be surrounded by several layers of cells containing 
sources of multi-access interference (i.e. other DS CDMA users). The interference will be 
subjected to multipath fading and shadow fading. Shadow fading occurs when structures 
(such as buildings, hills or mountains) attenuate propagating signals. Shadow fading varies 
more slowly than the multipath fading component of the signal and is interpreted as the 
time-varying mean of the rapid, multipath-induced signal fluctuations.^* 

The incident signal model may be modified sUghtly: 



K 

1 

m=l 



y{t) = AZIO'-'^" • c„(r - t„)b„{t - tJu„ + 



f^nsj20\ (5.2) 



V./20 



V ''".<> J 



where the first summation is for center-cell and the second summation (with index n) 
results from the outer-cell interference. All users have lognormally distributed power 
control error where em is N(0,OpJ). The quantity Noc is the number of outer cells while K 
remains the number of users/cell. The variable Sn is N(0,64) which in mm specifies 
lognormally distributed shadow fading with 8 dB standard deviation. The quantity rn.o* is 
the distance-dependent, fourth-order propagation loss between the n'th interferer and the 
center cell base station. The phase-shift vectors u^ and u„ are, as in the previous 
subsection of this chapter, composed of i.i.d. random variables which are N(0,1). Note that 
the incident amplitude A„ is an indexed quantity unlike the first summation representing 
the center-cell users. This is because the outer-cell amplitudes are determined by a hand- 
off to the outer-cell base station with the least path loss. This will be discussed in more 
detail shorUy. The quantity n(t) is a vector of complex AWGN with power o^. 



73 




Figure 5. 1 Spatial region for simulation of outer-cell 
interference. 



Figure 5.6 below shows a diagram of the arrangement of the hexagonal cells, each 
with a unit radius ( cell area = 3v^/2 ). The region of interest is circular and contains the 
equivalent area of Noc = 21.67 cells, excluding the center cell. The small circle at the 
center of each cell designates the position of each cell's base station. The wedge-shaped 
region within die larger circle contains the equivalent area of 3.61 cells and it is over this 
region that the spatial distribution of outer cell interferers is uniform, excluding the portion 
containing the center cell. The interference from this wedge-shaped region will be 
determined for each scenario and the results replicated to generate the interference 
components for the remainder of the circular region. 

The path loss which occurs between an outer-cell user and an outer-cell base station 
contains a deterministic propagation loss (« r'' ) as well as lognormally distributed 



74 

shadowing. The random components complicate the hand-off - or membership - of a user 
to a cell: a user is not necessarily serviced by the closest base station. Viterbi et al.'" 
investigated the properties of outer cell interference for the case of perfect power control 
and lognormal shadowing while Lee et al."^ investigated the effects of imperfect power 
control and lognormal shadowing. These previous works examined the case of up to 4 
base stations involved in the hand-off. Increasing the number of base stations in the hand- 
off beyond iVs = 4 will only slighfly decrease the outer cell interference for a given degree 
of shadowing and power control error. 

The hand-off was computed by selecting the base station path with the minimum 
base-station-to-mobile path loss. For the mth outer-cell user, this results minimizing a 
convenient ratio of path losses between the outer cell mobile and the outer cell base 
station and the outer cell mobile and the center-cell base station: 



mm 



' 






»-.. 








lO' 


10 








*-,. 




«„ 


A. 


•10 


10 


•10" 


To 



where: m = L..KN„ (5.3) 



where m represents the index over the outer-cell users in the wedge-shaped region and n is 
the index over the //b =11 base stations denoted by the solid black circles in Figure 5.6 
above. The quantities r^^ and 10'--"^'° are the propagation loss and lognormal shadowing 
respectively between the mth mobile and the nth base station. The quantities r^^ , 
10"*-" " and are the propagation loss and shadowing between the mth mobile and the 
center cell at the origin. The quantity 10"'-'"° is the power control error of the mth user 
with respect to the base station chosen for hand-off. As the figure shows, base stations just 



75 

outside of the wedge-shaped region are utilized in the hand-off calculations for the outer 
cell interferers. The shadowing and power control error components are assumed to be 
independent from path-to-path. Some authors"^'""* maintain that spatial correlations exist 
in shadowing components, but those effects are not incorporated into this model. The 
incident levels (A„'s of equation (5.3)) resulting from hand-offs in the wedge-shaped 
region of Figure 5.6 were replicated to generate interference levels for the remaining 
portions of the large circular region. The model used in this research agrees precisely with 
the results of Viterbi et al.'" which reported that for Nb = 3 the ratio of the outer cell to 
inner cell interference is 0.57. 

It was assumed that power control error correlates perfectly between hand-off base 
stations and does not enter into interference calculations until hand-off is chosen based on 
minimum path loss. This is in contrast to the work of Lee et al."^ which assumed closed- 
loop power control error was uncorrelated between hand-off base stations. The authors 
minimized the quantity: 



mm 

4,= 

n = I...Nb 







V. 




«-.. 


m,n 


•lO" 


10 


•lO" 


10 








V. 






c 


•lO" 


10 





where: m = L..KNg (5.4) 



where 10 '° is the power control error between the mth user and the nth hand-off base 
station. They assumed that Nb =3; hand-off occurred to one of the three closest base 
stations. 



76^ 

Analysis 

This section will briefly revisit the analytical results of chapter four which gave an 
asymptotic expression for capacity when signals were directional. Another analytical 
model will then be introduced which exploits the assumption of lognormal interferers. 

The expression given in equation (4.3) for the {Eh/No) eg may be rewritten slightly 
using equation (4.4) for a single cell: 






N.SNRifip.W^^' ^^^^ 






where 5A«/„ =(A^/ct^)-10'-^'° is the input SNR and Gp„ =|w>„| /||wj| is the 

normalized array gain towards the mth signal. In chapter four some assumptions (via 
equation (4.4)) allowed the sum term in the denominator of the expression for (Eb/No)^ 
to simplify into products of average terms, some of which are not explicit functions of the 
input parameters (i.e. avg{Gpm } = Gpi, m ^t 1, the interference signal gain). In addition, 
the gain towards the desired user Gpi = Gpd was modeled by its upper bound A^, the 
number of array elements. While these approximations were not accurate when considered 
separately, their quotient resulted in a convenient form and gave acceptable results ( see 
Figures 4.2 and 4.3). The per-element capacity was the slope of equation (4.8): 



„ 3NN, 



' ln(10)Q-'(Pw) inilO , ' 

10 '^ 200 '^ 



(5.6) 



where the term containing the Q-function is dominant for the range of power control error 
considered here. The term - analagous to a single-pole roll-off - resulted from modeling 



77 

the multi-access interference as an averaged quantity. The averaging operation left the 
desired signal as the only random variable and resulted in a lognormal distribution of 
(Eb/No)eff in which the standard deviation (in dB) is determined solely by the power 
control error standard deviation Opce (see equations (4.5), (4.6)). Under this simplified 
analytical model, the number of users K and the number of array elements N affect the 
mean of (Eb/No)eff but not the standard deviation. 

This series of assumptions gave acceptable results for the asymptotic case of strong 
nominal input levels if the signals were directional and had a moderate degree of power 
control error. With the addition of multipath fading the results of chapter four are less 
precise. The roll-off of a diversity combiner's per-element capacity is slower for low 
values of power control error because the standard deviation of (Eb/No)eff is not 
determined solely by the power control error Opce. 

A different analysis technique which might approximate the standard deviation of 
(Eb/No)eff more accuately has been examined by other authors. The model assumes that a 
sum of lognormally distributed random variables is also lognormal. Along this line, 
Schwartz and Yeh developed an iterative version of Wilkinson's method"^ and then used 
their model to evaluate the outage probability of a multi-cell AMPS system."^ Beaulieu et 
al. examined several methods for approximating a sum of lognormal random variables and 
concluded that the "best" choice of model depends upon the system parameters (i.e. the 
degree of shadowing and the magnitude of the outage probability). 

This research will supplement the analysis of previous sections by approximating the 
multi-access interference as a sum of lognormal r.v.'s. This method will result in a closed- 



78 

form expression for outage probability. An explicit equation for the per-element capacity 
will not be possible. 

If the array gain towards the interference may be modeled by a constant, the resulting 
noise and multi-access interference is approximated by: 

/j^ = yio'° +^^yio'°+ — 2 — (5,7) 

±i Gp; ± IGp.SNRi 

where the left-most summation represents the inner-ceU interference and the second 
summation represents the outer-cell interference. The right-most term results from the 
presence of noise. The outer-cell interference has been modelled as a quantity normaUzed 
by the inner-cell interference: the outer-cell summation is over K rather than KNoc. This 
approach has been used for incident signals by previous authors"'"^ and will be extended 
to the diversity array output via the gain constant F. 

Wilkinson's method begins with the assumption that the multi-access interference is a 
lognormal r.v. The first two moments of the Ima are then matched to the first two 
moments of the sums of lognormal random variables. The mean and second moment oil ma 
are given in equation (5.8) below: 



e[u] = e 



expj^hidO)— 



= exp{Pm,+P'(jl/2) 



F / , \ 3 



=(*:-.)exp(/J'.J/2).-^exp(/J'.,V2).^^^^ = *. 



79 



eIiL] = e 



expl21n(10)— 



= exp(2pni2 + 2p^ol) 



= E 



;iexpK) + ^iexpfe)^^^|^^ 



EllL] = 



(K - l)exp(2P'G^) + iK- m - 2)txp{pW, ) 



r p\^ 



—-\(Kcxp(2p'ai)+KiK-l)cxp(pW^)) 



Gp, 



+ 2 



Gp, 



GpiSNRi 



/i:(/^-l)exp(/3^(c7,^+(j,^)/2) + | 



{K-l)exp{P'G'j2) 



^2 



2GpSNRi 



3F 



GpfSNRi 



Kcxp(p'al/2) 



= b, (5.8) 



where the variable z is N(m, , a/> and )9 = ln(10)/10. Note that the exponential terms of 
the inner and outer cell interference are specified as N(0,cf^) and N(0,Gy^) respectively. 
The logarithms of the moments allow linear solutions of m^ and (5^ in terms of the 
moments of the sums {bubz): 



'«z=21n(Z.,)-iln(feJ 



CT^' =ln(Z;2)-21n(Z?i) 
Once the interference moments are determined then (Eb/No)effmay be expressed as: 



(5.9) 



80 



(e ^ 

y^oJeff 






x,-101og(exp(l))c 



= ^N. 



10 



10 



(5.10) 



and the outage probability is given by equation (5.1 1) below where log(») denotes the 
base- 10 logarithm. This method does not allow an expUcit closed-form expression for the 
per-element capacity. It will, however, allow that quantity to be extracted from outage 
probability contours as was done for the simulated results. The expression for the outage 
probability from the approximated moments: 



Pr 



r T^ \ 



. N 

\\"<>-'eff.df 



where: 
77 = 10*log 



<^B 



= Q 



r/-4e-101og(exp(l)H 



^3„Gp/ 



V^ 



Gp, 



= 10* log 



2 'Gp, 



G,=^al + \m\og\tx^(l))al 



(5.11) 



Results 

Most of the details of the simulation procedure are outlined in the section Simulations 
and Results of Chapter 4 with one difference: the faded, incident signals were no longer 
directional and a DOA was not specified. Each signal was specified by its power control 
error (and shadowing for the outer cell interferers), and its phase-shift vector which 
consisted of complex, i.i.d. unit-variance Gaussian random variables. For each trial the 
(Eb/No)eff was determined via equations (1.8), (1.10),(5.3) and (5.5) and was then 
compared to an outage threshold of 7 dB. A sample mean was tabulated over the trials 
for each combination of users and array elements. Even with fading the resulting outage 



81 

contours were roughly linear in N with slopes which decrease with increasing power 
control error. 

Figure 5.2 shows Pr((Eb/No)eff < 7 dB) versus the number of users with the power 
control error equal to 4 dB and with no outer cell interference. As in previous plots, the 
curves are plotted parametrically. The different curves represent outage probability for 
different numbers array elements N. For this value of power control error the performance 
is not drastically different than for the nonfaded case shown in Figure 4.4. The nominal 
input signal levels (without power control error) are such that Eb/No = 7 ( or SNRi = 7/Nc 
= 7/127 = -12.6 dB) for a single signal incident on a single array element feeding a 




Figure 5.2 Outage probability versus the number of incident signals with power control 
equal to 4 dB. The number of array elements A^ is a parameter. 



82 

conventional detector. When comparing Figures 4.4 and 5.2 it is easy to see the faded and 
unfaded systems have roughly the same outage performance. This similarity ceases for 
other values of power control error. 

Figure 5.3 shows simulated outage contours as the power control error varied. The 
top two plots show contours of Pr((Eb/No)eff < 7 dB)= 0.02 as the power control error 
varied from to 7 dB with no outer cell interference. The upper plot (a) is for a nominal 
input condition (without power control error) of SNRi = 7/Nc = -12.6 dB while plot (b) 
shows contours for a nominal input condition of SNRi = lO^/Nc = 59 dB. Plots (c) and (d) 
show the same conditions but include outer cell interference. The plots show that, even 
with fading, outage-based capacity continues to be linear in ^V. Note also that as the 
nominal input levels increase, the intercepts of the approximately linear contours move 
towards the origin. This general behavior was predicted for the nonfaded case by equation 
(4.8). 

The slopes of the lines in Figure 5.3 were extracted using a least-squares curve fit and 
plotted as the per-element capacity in Figure 5.4. Note that tiie per-element capacity is less 
than that for the case of no Rayleigh fading as long as the power control error is less tiian 
approximately 1.5 dB. As long as tiie power control error is greater than 1.5 dB the 
receiver employing a diversity array combiner in the presence of fading outperforms a 
receiver using a beamformer to equaUze signals which are not subjected to multipaUi 
fading. 



83 




15 
(a) 

T- 



4 
(b) 



20 25 30 

Number of Array Elements 




5 6 7 

Number of Array Elements 




5 6 7 

Number of Array Elements 

Figure 5.3 Capacity versus the number of array elements with power control error 
as a parameter. Outage probability = 0.02. Plots (a), (b) are without outer cell 
interference. Plots (c), (d) include outer cell interference. 



84 

Note that in Figure 5.4 there is degradation in the per-element capacity as the nominal 
input levels increase and outer cell interference is present (plot (b)). When the nominal 
signal level is small (SNRi = -12.6 dB) the individual outer cell signals incident on the 
inner cell are very weak (compared to ambient noise and the inner cell interferers) due to 
the path loss. In this circumstance the outer cell interference probably just adds to the 
ambient AWGN. When the nominal input signals are large {SNRi = 59 dB), outer cell 
interferers can overcome the outer-cell to center-cell path loss and can have signal levels 
much higher than the ambient noise, even at the center cell. In this case some percentage 
of individual outer cell signals compete with the inner cell interference for attention from 
the array processor, and performance suffers. 



Per- 

Element'' 

Capacity 



Per- 



15 



Element 
Capacity 



10 



1 I 

e O — G SNRi = -12.6 dB 

SNRi = 59.6 dB 




1 


I 


1 r- 

i ^ 
• .• 


-G- 


-9 SNRi = 


= -12.6dB 
= 59.6dB 


"""""-"!•"- 


~~-S^ 


• 




— SNRi = 


i 1 


^^^^^f-^-i^-^^^ 



3 4 5 6 7 

0^) Power Control Error (dB) 



Figure 5.4 Per-element capacity (users/array element) versus power control error 
(dB) for an outage probability = 0.02. The upper plot is for the case of no outer 
cell interference. The lower plot includes outer cell interference. 



85 

Figure 5.5 below shows the per-element capacity from the simulated results of Figure 
5.4b (with outer cell interference) and the approximation given in equation (5.2) with ^ = 
7 dB and Gpi = 2. Agreement between (5.2) and the simulated results for the case no 
outer cell interference was poor, and the results are not presented here. The value of Gpt 
accounts for the fact that the output power of the outer cell interference is about equal to 
the output power of the inner cell interference when averaged over most of the possible 
combinations of user population, array elements and power control error. The rapid roll- 
off of the curve from equation (5.2) is due to the use of averaged multi-access interference 
when computing (Eb/No)eff, as discussed in a previous section. 

Figure 5.6 shows the per-element capacity resulting from the application of 
Wilkinson's method to model the multi-access interference as a sum of lognormal 
variables (equations (5.7) through 5.(1 1)). The lognormal statistics of the inner cell 
interferers are assumed to be due to power control error only while the outer cell 
interference combines power control error and 8 dB of shadowing {xm of equation (5.7) is 
N(0, Opce^)) while y„ is N(0, q^ + 64)). Simulations show that the average interference 
gain Gpi for the inner cell interferers can vary from zero to imity depending on the 
number of elements, the number of users and the nominal input level. Interestingly enough, 
Gpi does not seem sensitive to the power control error. Averaging over these conditions 
results in Gpi » 0.7. The gain constant for the outer cell interferers - F of equation (5.7) - 
is taken to be unity. Simulations have shown that the array attenuates the interference so 
that the average output power due to the inner and outer cell interferers is about equal 



86 



£.\) 




' 


' 


1 1 


1 1 


18 


-'•. 






•f ♦•• 


• Eqn. (5.2) 


16 
Per 


• 
* 


'•. 




-© 9- 


Simulations, 
SNRi= -12.6dB 


Element14 
C^)acity C 


;^ 


• 
V. 

• 






Simulations, 
SNRi= 59 dB 


12 


■ 








■ 


10 
8 
6 


"^ 






"^V 


• 


4 


• 






^^^^^S^ 


• 


2 


• 








^*'*'^^**t>^^ 


n 




1 






^^^"^^^^^^^n 





1 


2 


3 4 


5 6 7 










Power Control Error (dB) 



Figure 5.5 Per-element capacity versus power control error (dB) with outer cell 
interference. The capacity is from equation (5.2). 

over many scenarios of interest. Interestingly enough, simulations show the ratio of the 
incident signal power from the outer cell and inner cell sources is 0.57. 



The upper two plots of Figure 5.6 show that Wilkinson's method does not provide a 
particularly useful approximation when there is no outer cell interference. The lower two 
plots include the effects of outer cell interference and tend to agree more closely with 
simulated results for a range of power control error from 1 to 6 dB. For power control 
error = 7 dB the simulated per-element capacity is almost an order of magnitude greater 
than the analytical results. 



87 



Per 30 
Element 
Capacity 
20 



10- 



1 



Per 30 
Element 
C^jacity 
20 



10 



■I 1 1" 1 

^>.^. .:. ; ; 


I 1 

SNRi = -12.6 dB 
Analysis 


^~^J i i 








i i i 





4 
(a) 



1 

i , 


1 

-: 

^...'^i.^. 


■]■■■■ 


I 


SNRi = 59 dB 


" Analysis 






i 


t~ 


•^^^TtQ:^ 


J ' 


^ ^ 



4 

(b) 



Per 15 
Element 

Capacity 

10 



Per 

Element 

Capacity 




5 6 7 

Power Control Error (dB) 



Figure 5.6 Per-element capacity versus power control error (dB) for outage 
probability = 0.02. Plots (a) and (b) are without outer cell interference. Plots (c) 
and (d) include outer cell interference. Curves from analysis result from equations 
(5.8) -(5.11). 



88 

Conclusions/Discussion 

Simulations showed that even in the presence of multipath fading, the per-element 
outage-based capacity might serve as a useful performance measure for a wide range of 
conditions when considering a single-cell with multi-access interferers. As the nominal 
input levels increase for the multiple-cell case the per-element capacity deteriorates for pee 
^ 3 dB compared to the single-cell case. The reason: for low nominal input levels the array 
treats the outer cell signals like AWGN. When the nominal input levels are high all 
interferers incident on the base station are well above the noise and the array must 
r dedicate some processing to attenuate the outer cell interferers, at a loss in performance. 

The simulated results presented in this chapter show that for low nominal input levels 
(SNRi = 7/127 = -12.90 dB) and power control error > 1.5 dB a single-cell system with 
multipath faded signals will have a higher per-element capacity than a system with 

directional, unfaded signals. The addition of outer-cell cochannel interference 
degrades the per-element capacity by about 80-90% for p.c.e. < 2 dB and 60-70% for 
p.c.e. > 2 dB. For strong nominal input levels {SNRi = 59.6 dB) the addition of outer cell 
interference degrades the per-element capacity by a factor of 2/3 to 3/4 for power control 
error < 2.5 dB. 

The two analytical models for per-element capacity gave mixed results. The model 
developed in Chapter 4 - which averaged multi-access interference - degrades too rapidly 
with increasing power conttol error to be of much use in a single-cell scenario. In a multi- 
cell scenario witii low input levels the model gives a good fit to simulated results for 
power control error > 1 dB. When input levels are high, tiie array output power due to tiie 



89 

outer cell interferers can be as much as twice the power of the inner cell interferers (there 
are 21.67 times more signals originating in the outer cells than the inner cells). This 
accounts for the decrease in the capacity in Figure 5.5 as the nominal input levels increase. 

This model is limited in several ways. First, we are modeling the interference as an 
averaged quantity so the complex trial-to-trial interactions between the interference, the 
desired signal and the noise are lost. Second, we are attempting to absorb the relatively 
complicated behavior of the processor into two parameters, Gpi.and N, which do not vary 
with the number of users or the power control scenario. 

The model based on Wilkinson's method of evaluating sums of lognormal random 
variables was introduced in an attempt to resolve the first of these two issues. It was 
hoped that the accuracy of the analytical model might be improved by approximating the 
interference as a sum of variables rather than an average. The plots in Figure 5.6 show that 
this model might offer some improvement in accuracy - compared to simulations - for the 
select case of multiple cells. If more accuracy is required for the case of a single-cell 
system or power control > 6 dB, then refinements of the models would be necessary. 

Summary 
This chapter presented simulation results of the outage-based capacity for incident 
signals subjected to frequency-nonselective Rayleigh multipath fading and lognormally 
distributed power control error. The models included single-cell and multi-cell scenarios 
where outer cell interferers were subjected to lognormal shadow fading. The multipath 
fading model assumed that the spatial interactions between the faded incident signals and 
the antenna array allowed the fading process to be fully decorrelated between array 



90 

elements. Under these conditions the array functioned as a diversity combiner rather than a 
beamforming antenna array. 

The per-element capacity - the number of users/array element which may be 
supported for a required outage probability - was evaluated via simulations. The 
motivation behind the use of per-element capacity was to find a performance measure 
which was robust to the variations in user population and array size and which would 
reflect the possible contributions a receiving array would make to the capacity of a power- 
controlled CDMA system.. An additional goal was to formulate approximations which 
would allow easy assessments of the per-element capacity. 

Physical arguments in conjunction with semi-empirical curve fits resulted in two 
analytical models. The first model was adapted from the model in Chapter 4 in which the 
signal, the interference and the noise were averaged. The complex interactions between 
even these quantities was overlooked for the sake of simpUcity. Array outputs were 
modeled as averages of input quantities and gain constants. Values of these constants were 
derived empirically from simulations. 

The second model exploited Wilkinson's method of approximating lognormal 
variables so that averaging of the interference could be avoided. This model resulted in 
sUghtly improved accuracy - compared to the first model - for power control error < 2 dB 
but was somewhat worse for power control error > 6 dB. 



CHAPTER 6 
BASE STATION ANTENNA ARRAY ADAPTIVE PERFORMANCE 

This chapter examines the steady-state performance of the Recursive Least Squares 
(RLS) adaptive algorithms for a base station diversity array receiving faded incident 
signals with power control error. A simulation approach is used since the scenarios are too 
complicated to allow useful analytical solutions. The simulations were discrete-event 
simulations of transmitted bits through a nonstationary AWGN channel with time-varying 
multipath fading as well as stationary power control error and shadow fading. The 
simulation results will be given in terms of histograms of the effective Eb/No which can 
facilitate outage calculations. The performance of an RLS array in the presence of 
cochannel interference has been examined by several authors,^^-^^" but of these only 
Tsoulos et al. considered near/far scenarios with fading and outer cell interference. They 
used simulations to determine the outage probability when the (A^ = 8) array functioned as 
a beamformer, not a diversity combiner. 

Simulation results show that the RLS algorithm can track the time-varying solution 
effectively and that the adaptive solution is close to the steady-state solution. The 
structure of this chapter is similar to the two previous chapters. The first section revisits 
the now-familiar signal model from previous chapters and also introduces the time- 
dependent fading process. Approximations of the fading process used in this research are 
also presented. The second section outlines the RLS adaptive algorithms and the receiver 



91 



92 

structure used in the simulations. The third section gives the results of the discrete-event 
simulations and provides some discussion. The last section gives the conclusions. 

Si gnal and Channel Model 

The signal model is the same as in the previous chapter: 

i;W = AilO'"-.c„(r-rJfc„(r-Tju„-h 

IA„.10'"-. 10_ .c,(r-T,K(r-r„K+n(r) 

where the first summation is for center-cell and the second summation (with index n) 
results from the outer-cell interference when it is taken into account. Power control error 
and shadow fading are lognormally distributed {e^ '\&N(0,apJ),Sx is N(0,64)). The 
quantity Noc is the number of outer cells while K remains the number of users/cell. The 
quantity r„ is the distance-dependent, fourth-order propagation loss between the nth 
interferer and the center cell base station. As in Chapter five the amplitude A„ accounts 
for the hand-off with the least path loss when outer cell interference is taken into account. 
The quantities c^^ )an<ibx() represent the spreading code and the BPSK modulation with 
ideal, square pulses. We assume the beginning and end of a FN sequence corresponds to 
the beginning and end of an information bit. In this chapter the FN sequences, no longer 
random, are Gold codes of length 127. The quantity n(t) is a vector of complex AWGN 
with power a„^ which is temporally and spatially white. The power control error and 
shadow fading are assumed to be constant over the observed interval. 



93 

Time-varying, flat Rayleigh fading, is introduced into the incident signals by the 
phase-shift vectors Un, and Un. The fading process is independent between array elements 
(spatial samples) but is correlated between time samples. The time autocorrelation 
function is given by: 

R{T) = JoicOoT) (6.2) 

where ttb is the Doppler frequency in radians/sec, x is a time delay and Jo(*) is a Bessel 
fimction of the first kind with order zero. The Doppler frequency arises from relative 
motion between the signal source and the observation point; it represents a velocity- 
induced shift in the signal carrier frequency. The Doppler frequency is given by ttb = ^ 
where k is the free-space wave number (k = liWk ) and v is the relative velocity. 

The fading of the incident signals was accomplished by superimposing a time- varying 
complex Gaussian component on the time-dependent data waveforms. The fading 
component was generated by a third-order autoregressive (AR) filter. The filter response 
to white noise has an autocorrelation function with a damped sinusoid characteristic which 
has been adjusted to closely match the major lobe and first minor lobe of the Bessel 
function in equation (6.2). The impulse response of the filter is given by:*" 

M0 = (^^''^*''[2.66/oSin(5.15/o0-3.17/oCOs(5.15/or)-f-3.17e-"'^'''])M(r) (6.3) 

where /d is the Doppler frequency in hertz and u(t) is a step function at r = 0. The filter is 
used in the following manner. First, complex white Gaussian noise is processed with the 
filter specified in equation (6.3). The data waveform is then multiplied point-by-point by 



94 

the filter ou^ut to simulate the fading process. This approach preserves the salient 
properties of the fading and allows an efficient simulation procedure. 

Adaptive Receivers 

The RLS processor has an advantage over the LMS algorithm in that its convergence 
rate is not dependent upon the eigenvalue spreads of R, the input autocorrelation matrix. 
The disadvantage of this algorithm is that the computational burden (0(N^)) is relatively 
large compared to the computationally efficient Least-Mean-Squares (LMS) algorithm. 
Unfortunately an LMS array cannot track the time-dependent fading, especially at higher 
vehicle speeds. "^^ 



processor 1 




Figure 6.1 Adaptive array with a discrete-time conventional detector. 



95 



The discrete-time adaptive receiver structure for user 1 is shown in Figure 6.1 below. 
The outputs of the antenna elements are passed through integrate-and-dump filters 
matched to the chip transitions of the desired signal waveform. The sensor array outputs 
are then weighted and summed. The array output is passed through a discrete-time version 
of a conventional detector which consists of a delay-Une filter with Nc discrete time-delay 
elements each with a duration of one chip (dotted box). Once every bit the contents of the 
delay line are weighted by the elements of the PN sequence (Ci = [ci Ci Ci ...Ci " ] e 
{ 1,-1 }) of the desired signal and summed to form the output The estimate of the current 
output bit is taken as bi,est(n) = sign(Real{So,i(n)}). The array weights are updated once a 
bit For the nth bit interval the input of the array and the weights are given by: 

w(n) = [wo{n) w,(/j) • • • w^_,(/i)f 



X(«) = 



x°{n) xl{n) 



(6.4) 



^:- 


■'(")■ 


^r- 


■\n) 


■*Af-l 


"(«). 



where Xg'"(n) represents the array input at tiie gUi element during tiie mth chip interval of 
the nth bit. The inputs are processed by tiie integrate-and-dump filters. The input to tiie 
bank of weights is given by: 



Y{n) = 



nT 



hT-Tc 

\xl{n)dt \xl{n)dt 

nT-2Tc 



hT-T, 



hT 



nT-(N,-\)Tc 

x:^-\n)dt 

hT-NcTc 






nT-Tc 

jxl_,{n)dt jxl,_,{n)dt 

nT-7Tc 



'x',i;'{n)dt 



nT-T, 






hT-N^T^ 



(6.5) 



The inputs are weighted and summed by tiie array and tiie resulting output stream is 



96 

weighted and summed by the tapped delay-line. The array/detector attempts to minimize 

the output error: 

ein) = \Un) - w'(n)Y(n)c, = ft,„,(n) - w^(n)z(/j) 

(6.6) 

where the random variable is z(n) = y(n)ci. A single element of z(n) is the inner product 
between the output of one array element - filtered chip-by-chip over the bit interval - and 
the desired user's PN sequence Ci. The vector z(n) will be the sample vector over which 
the adaptive filter operates. 

The RLS algorithm recursively computes an estimate of the optimum weight vector 

Wo. The recursive equations are given by (version 2, Haykin.^^): 

^ y-'R-'(n-l)z(;z) 
^ ^ l + r'z"{n)R-\n-l)z{n) 

a{n) = r{n)-w"{n-l)z{n) 

(6.7) 
w(n) = w(/i - 1) -I- k{n)a'{n) 

R-i(n) = y-'R-'{n - 1) - y-'k{n)z" {n)R-\n - 1) 

where a(n) is the a priori estimation error and is an estimate of the error between the 
current value of the reference signal r(n) and the output based on the previous value of the 
weight vector at iteration n-1. The weight vector is updated by the product of k(n) and the 
a priori estimation error. The vector k(n) may therefor be interpreted as a gain vector and 
is sometimes called the Kalman gain. Note that the matrix inverse is updated recursively, 
rather than by a block computation. 



97 

The quantity y serves to exponentially window the data. Eleftheriou and Falconer"* 
investigated the influence of y in the adaptation process. They examined the role of y on 
the total misadjustment, the sum of the "estimation noise," which results in the 
displacement of the filter weight vector from the optimum value, and the "lag error" which 
occurs when the filter tries to track channel nonstationarities. Too large a value of y will 
cause a filter to use outdated statistics when tracking channel nonstationarities. An 
excessively small value of y will allow fast adaptation but will cause too little information 
to be used in the adaptation process and result in excessive misadjustment In this work 
the selection of y was empirical. More details will be given in the next section. 

In the adaptive receiver shown in Figure 6.1 the bit estimate besi(n) serves as the 
reference signal r(n) for the RLS processor. The structure shown is for decision-directed 
operation only. Training and transient response issues as well as issues arising from fixed- 
point arithmetic were not addressed by this research. 

Simulations 

Monte Carlo simulations were conducted for the discrete-event case. Rather than 
examining exhaustive combinations of power control error, user populations and array 
sizes, discrete-event simulations focused on a few combinations of these quantities in an 
effort to show that the adaptive solution does not differ appreciably from the steady-state 
results presented in an earUer chapter. 

Results in this chapter are presented in terms of the histograms of the (Eb/No)eff.dB 
When (Eb/No)eff is tabulated in dB units for many trials the resulting histogram is 



98 

approximately Gaussian but contains some slight asymmetries. This result is not surprising 
considering that the mean incident signal amplitudes are lognormally distributed."'"*^ 
Some authors have found that a sum of lognormally distributed random variables is well- 
approximated by another lognormally distributed random variable. 

All signal sources were assumed to have velocities of 55 mph relative to the base 
station. This corresponds to a Doppler frequency of 74 Hz. A real mobile communication 
system must cope with multiple signals Doppler-shifted over a range of values: zero for a 
car sitting still or 1 14 Hz for a driver going 85 mph. Note that the Doppler frequency or 
frequencies could have an appreciable impact on the selection of Yr, the RLS memory 
factor. 

Incident signal levels were modeled via the signal description given earlier in the 
chapter (equation (6.1)). All incident signals consisted of random, equally-likely BPSK bits 
(rate = 9600 bits/sec) which were spread with Gold codes of length 127 and chip-filtered 
via the integrate-and-dump filters of Figure 6.1. The effects of multipath fading were 
introduced via Um and Un - the independent array samples - in equation (6. 1) and the 
multipath fading simulator described in an earUer section. The fading was generated at the 
chip rate and was not considered constant over a bit interval. A single scenario consisted 
of multiple signal vectors of 51 bits in which power control error and shadow fading were 
constants and only the multipath fading component of the incident signals varied with 
time. 

The RLS processor was modeled via equation (6.4) above. In order to avoid 
transient-related issues the initial weight vector for the RLS algorithm was the steady-state 



99 

MMSE weight vector averaged over the first bit Output signals were then tabulated over 
the next 50-bit interval and (Eb/No)eff was calculated for each single bit. Six hundred 
scenarios for a total of 30,000 RLS-adapted bits were simulated. The nominal input SNRi 
= -12.6dB. 

The choice of y was made empirically. A scenario was considered with 5 inner-cell 




50 100 150 200 250 300 350 400 
Time in Bits 



Figure 6.2 The time-varying (Eb/No)eff for an RLS processor and an ideal MMSE 
processor. There are 5 users with pee = dB incident on a 3-eleraent array. The 
Doppler frequency is 73 Hz. and y is 0.825 

users with pee = dB subjected to multipath fading. The (Eb/No)eff was tabulated on a 

bit-by-bit basis for an interval of 400 bits using the output of an RLS processor 

((Eb/No)eff,RLs) and the output of an ideal MMSE i(Eb/No)eff.MMSE) processor. For these 

conditions a value of Xr = 0.82 was found to minimize the quantity: 



100 



I- J 400 „^i W No J ^,RLS \N0 Jeff MUSE J 



(6.5) 



and was used to produce the results that follow. Figure 6.2 shows the time-varying 
(Eb/No)eff for the ideal MMSE processor and the RLS processor over the 400-bit interval. 

The results of simulations for 5 users with Opce = dB is shown in Figure 6.3 below. 



0.14 
0.12 
0.1 
0.08 
0.06 
0.04 
0.02 h 



-30 



-20 -10 

Effective Eb/No in dB 









N = 3 
K = 5 
pce = OdB 
fD = 73Hz 
- Y= 0.825 


: :in= 10.98 dB 
i i o = 3.46 dB 


t 


\^ ... 




1 , 




: I 






i i 






I 1 






\ — 

i Iv 


— — 


1 i nfll 


mm L 



10 



20 



Figure 6.3 Bar chart of histogram of (Eb/No)eff in dB. The soUd line curve oudines 
a Gaussian pdf with mean = 10.98, st. dev. = 3.46. There are 5 users with pee = 
dB incident on a 3-element array. The Doppler frequency is 73 Hz. and y is 0.825. 

The plot shows a bar-chart histogram of (Eb/No)eff in dB. Superimposed on this is 

Gaussian curve with mean and standard deviation taken as sample means (m = 10.98, a = 

3.46) from the data which generated the histogram. The match between the curves is good 

except for the peak and the tail regions. Unfortunately, the lower tail region is critical for 



101 

outage calculations and the poor fit might prevent accurate results if a lognormally 
distributed output is assumed and the outage probability is low. 

Another interesting feature of Figure 6.3 is the slight asymmetry between the upper 
tail regions of the bar-chart curve. The asymmetry is also present in histograms generated 
by steady-state simulations. Using the histograms to calculate the outage probability gives 
Pr{(Eb/No)eff ^ 7 dB) = 0.105, .125 for the bar-chart and Gaussian curve respectively. 
Under the same conditions an ideal MMSE processor gave a mean output of (Eb/No)eff = 
12 dB, a standard deviation of 3.27 dB and Pr(,(Eb/No)eff ^ 7 dB) = 0.066. Figure 6.4 
shows the same conditions except that power control error has been increased to 4 dB. 
Note that the mean is only slightly decreased to 10.7 but the standard deviation is 
increased to 5.37. 

Histograms which include the effects of outer cell interference are shown below. 
Figure 6.5 shows the histogram of (Eb/No)eff for the case of perfect power control in 
which outer cell users are subjected to 8 dB of lognormal shadow fading. The assignment 
of the power levels for the outer cell users is the same as described in Chapter 5. As in the 
previous plots a Gaussian distribution with the same mean and variance as the bar chart 
histogram is shown. The mean ( = 10.6 dB) decreased sUghtly compared to the case of no 
outer cell interference (= 10.98 dB ). The standard deviation (= 3.33 dB) decreased 
slightly from a value of 3.46 dB for no outer cell interference. Using the empirical 
histogram results in an outage probability of Pv{(Eb/No)eff ^ 7 dB) = 0. 15, a sUght 
increase compared to the case of no outer cell interference. This seems to agree with the 
general trends noted in the steady-state results which show only a slight degradation in 



102 

capacity when outer cell interference is included. The case of pee = 4 dB is shown in 
Figure 6.6. The mean of the empirical bar-chart histogram is 9.75 dB and the standard 
deviation is 5.17 dB. Unlike the previous figures, these curves resulted from 10000 bits of 
data. 

Conclusions/Summary 

This chapter has examined via simulation the outage performance of an adaptive 
diversity antenna array updated with the RLS algorithm. The output performance was 
tabulated by histograms of (Eb/No)eff in dB. The adaptive performance did not vary 
drastically from the steady-state performance. It appears that the RLS algorithm operating 
in a decision-directed mode is capable of tracking the time-varying channel when Rayleigh 
multipath fading is present 



103 




Figure 6.4 Bar chart of histogram of (Eb/No)eff in dB. The solid line curve oudines 
a Gaussian pdf with mean = 10.98, st. dev. = 3.46. There are 5 users with pee = 4 
dB incident on a 3-element array. The Doppler frequency is 73 Hz. and y is 0.825. 



104 




-5 5 

Effective EbNo in dB 



15 20 



Figure 6.5 Bar chart of histogram of (Eb/No)eff in dB for the case of outer cell 
interference. The solid line curve outlines a Gaussian pdf with mean = 10.6, st. dev. 
= 3.33. There are 5 users with pee = dB incident on a 3-element array. The 
Doppler frequency is 73 Hz. and y is 0.825. 



lOS 



0.12 



0.1 



0.08 



0.06 



0.04 



0.02 - 



N = 3 

K = 5 users/cell 
■ pee = 4 dB 
fD = 73Hz. 
Y= 0.825 


1 -t 1 — 

• ■ ■ 
« • I 

• t • 

L 1 n.. 

m = 9.75 dB J 

: a = 5.17dB -\ 
I . . J II 


......... 










Vj 


[ 










i >> 










7, 


i 


t 


» 




yuSffifOulllll 


1 



-30 -20 -10 10 

Effective Eh/No in dB 



20 



30 



Figure 6.6 Bar chart of histogram of (Eb/No)eff in dB for the case of outer cell 
interference. The solid line curve outlines a Gaussian pdf with mean = 9.75, st. 
dev. = 5.17. There are 5 users with pee = 4 dB incident on a 3-element array. The 
Doppler frequency is 73 Hz. and y is 0.825. 



CHAPTER? 
SUMMARY 

In this work we have examined the possible performance contributions that adaptive 
antenna arrays can make to DS-CDMA base station performance. This final chapter will 
provide a results-oriented summary of the previous chapters. In addition, a section entitled 
Areas for Future Work will outline some of the issues which have arisen during this 
research which are related to array processing and channel modeling. 

The system model has consisted of an array of antennas whose weights are controlled 
by a MMSE processor. This work has examined two system configurations. First, a multi- 
user antenna array was investigated which consisted of an array of sensors and a single 
beamforming processor. The processor with a single set of weights attempted to equalize 
the multiple incident signals. The second structure used an array of sensors which was 
shared by multiple processors, each dedicated to a single incident signal. Each distinct 
output fed a DS conventional detector. 

Chapter two examined the steady-state performance of the multi-user processor. 
Unlike most analyses of MMSE filters, this analysis focused on output SNR as the 
performance measure. First, the case of K <N-1 spatially orthogonal users was examined, 
followed by the case of 2 incident signals with an unspecified DOA spacing. The analysis 
showed that for strong incident signals the output SNRs of the various signals were 
leveled to a value near that of the weakest user. 



106 



107 

Chapter three investigated the steady-state adaptive performance of an LMS multi- 
user processor and attempted to establish a link between the misadjustment of the LMS 
algorithm and the output SNR performance. The performance was determined by 
examining first two moments of the output SNR. The first performance measure was a 
cost function based of the difference between the mean SNR response of the processor 
and the steady-state output performance. An upper bound for the cost function was 
formulated which limited the excursion of the mean output SNR from the ideal steady- 
state value. The second cost function was the quotient of the output variance and the 
mean squared. Analysis and simulations showed that both cost functions were found to be 
held to acceptable limits if the step size was selected using the well-known rules based on 
convergence argimients of the LMS algorithm. 

Chapter four examined the steady-state performance of a single-user processor. The 
performance for this and the remaining chapters was judged on the ability of the array to 
recover the desired signal and reject multi-access interference. This chapter attempted to 
provide some insight into the contributions an array could make to the base station 
receiver performance when multi-access users limited their transmit power via imperfect 
power control. Rather than performing a rigorous analysis, an approximate analysis - 
which replaced the interference response with its average - resulted in simple expressions 
for the outage-based capacity. The outage-based capacity was found to be Unear in a 
quantity which was the product of the processing gain and the number of array elements. 
A robust performance measure - per-element capacity - was defined as the slope of this 
line with the processing gain held constant Analytical results were corroborated with 
exhaustive simulations. 



108 

The next chapter introduced multipath fading and outer cell interference into the 
steady-state channel model. The approximate analysis based on the techniques of chapter 
four gave mixed results. The approximate analytical model based on techniques from 
Chapter four did not provide a good fit to simulated results for the per-element capacity in 
the case of a single cell. The same model agreed well with simulated results for the 
multiple-cell case as long as the power control error exceeded 2 dB and the nominal input 
level was low (SNRi = -12 dB). Another model was introduced which approximated the 
array gain by an average. The incident signals themselves were modeled using Wilkinson's 
method of approximating sums of lognormal variables. The resulting expressions for the 
per-element capacity were more accurate than the models from chapter four, for the 
faded-signal case. When the asymptotic case of strong incident signals was considered, the 
analytical expressions for per-element capacity gave slightly optimistic results compared to 
simulations for power control error less than 4 dB. 

Chapter six examined the adaptive performance of the MMSE processor via 
simulation using the RLS algorithm. Performance was expressed through histograms of 
(Eb/No)eff,dB and Gaussian curves with same mean and variance as the simulated data. It 
was found that the mean and variance of the adaptive results agreed closely with those of 
the steady-state simulated results. Unfortunately, the histograms were slightly skewed, 
with the peak higher than the sample mean and a lower tail more pronounced than the 
upper tail. The ou^jut was not exactly lognormal and that the skew might prevent accurate 
analytical assessments of outage performance if the outage probability is small. 



109 

Areas for Future Work 

Two issues have arisen during the course of this research which might warrant further 
investigation. The first issue is the application and extension of previous woric to provide 
rigorous, useful measures of array performance, particularly when then number of users is 
large. The second is the extension of existing channel models to include the effects 
spatially-dependent multipath fading. 

This research provided semi-empirical approximations as convenient - but limited - 
measures of outage-based performance. A more rigorous analysis is warranted. The 
analysis might best focus on the joint statistics of the desired signal, the multi-access 
interference and the noise. One possible approach is to find the pdf of the SINR: 

SINR = s^'Q-'s* = tr(sVQ-' ) (7.1) 

where Q is the covariance matrix and s is the signal vector of the desired signal. K signals 
are subjected to independent Rayleigh fading then phase-shift vectors will consist of zero 
mean jointly Gaussian i.i.d. elements. The joint distribution of the multi-access interference 
components of Q are known to be Wishart distributed."' It might be possible to 
formulate a pdf using these statistical relationships. This problem was noted, but not 
solved, by Dlugos and Scholtz.'^" 

Using previous work resulting from investigations into the Sample Matrix Inverse 
processor (SMI) might provide a good starting point. The SMI processor uses a sample- 
mean estimate of the covariance matrix Q: 

1 '^ 
Q = t;Xx„x; (7.1) 



no 

to compute an estimate of the weights, w^ = Q"'s* , where x^ is the array output vector 

of jointly distributed complex i.i.d. N(0,ax) variables at the mth time sample. The vector s 
is the A^-length, phase-shift vector of the desired signal and consists of unit-amplitude, 
complex exponentials. Obviously M > N so that Q is full rank. Previous woric has 
examined the relationship between the window length M and functions and various 
functions of the output power for stationary'^''" and nonstationary noise.'" It has been 
shown that the output SINR normalized by its steady-state value is a random variable with 
a beta distribution'^' for the case of stationary noise. 

Reinterpreting this previous work may lead to some convenient results for diversity 
combining. Recall that the samples that led to Q were interpreted as samples of the 
Gaussian, zero-mean interference and noise. If the samples are instead phase-shift vectors 
of Rayleigh-faded multi-access interferers, then Q may still be interpreted as a sum of 
outer products of jointly Gaussian vectors. The results of some previous radar-oriented 
research might be reinterpreted in terms of diversity combining. If a meaningful, 
unambiguous measure of the output SINR of an optimum combining array is possible, 
based on previous results, it might provide a robust solution to the problem of calculating 
array performance when the number of users exceeds the number of array elements. This 
is currently an active area of research. '^'*'^' 

The second area of future work is the extension of general channel models to include 
the effects of spatially-dependent multipath fading. A great deal of work has been done in 
the area of temporal single-channel fading models but relatively few models exist which 
include spatial effects for multi-channel receivers. The interest in antenna arrays as a 



Ill 

component in commercial communication systems has grown quickly in recent years. 
Unfortunately, developments in array theory have outpaced the channel models. For 
antenna arrays to reach their full potential as a communication system component 
accurate, useful models must be developed which provide a unified, integrated description 
of the temporal and spatial characteristics of the channel. 

One such model is an extension of the single-channel with multipath fading which 
assumes that multiple plane waves originate from the same source but arrive at a point in 
space via different paths. In the single-channel model the effects of the individual waves 
are modeled by "rays" which result in a Rayleigh-distributed amplitude and can result in 
relative time delays between same-source arrivals. This model has been crudely extended 
to include spatial effects by assuming individually faded, dominant rays arrive at points in 
space from different directions. This type of model focuses on the distribution of large, 
dominant reflectors in the environment about the receiver but ignores the smaller-scale 
spatial effects at the receiver itself. This is the basis of ray tracing techniques'^*'^^ which 
use site-specific measurements to predict path loss. Even current ray-tracing models focus 
on single-channel measurements of power-delay profiles. This gives a macroscopic 
description of the transmission amplitude versus delay but ignores local spatial 
characteristics at the receiver. 

Other models have also been proposed which describe the impact of the spatial 
channel locally at the receiver but do not consider the surrounding environment. Salz and 
Winters'"' examined the effects of a continuum of wavefronts with a nonzero DOA spread 
on the crosscorrelations between array elements. As an example, the simple fading model 
used in this research assumes the crosscorrelations are zero. Raleigh et al."" have also 



112 

proposed a time/space model to describe the effects of multipath on the spatial samples 
from an array. There is plenty of room for additional local models which incorporate 
space/time descriptions for an array-based receiver. 

These models seem like a first tentative step towards a unified channel description. 
Local receiver models need to be integrated into the larger-scale models of the 
surrounding environment. This may dictate new, or at least more detailed, descriptions of 
the processes which comprise the unified time/space channel. 



APPENDIX A 
THE OUTPUT S>fR PERFORMANCE SURFACE 

This appendix gives some of the general aspects of the output SNR as a performance 
surface. There are some important differences between the MSE and the SNR 
performance surfaces. Some general qualities of the surface in weight-space are noted, 
such as the contours and the gradient, along with the impact of misadjustment on the 
multi-user processor. 

The best way to get an intuitive feel for the SNR performance of an MMSE processor 
is to examine the case of a single user in additive white Gaussian noise. The weight-space 
diagram for a single-user case is shown in the figure below. The elliptical curves are the 
customary constant MSE contours and the straight lines represent contours of constant 
output SNR. Notice that the constant SNR contours are lines which pass through the 
origin and only the slopes change with changing SNRo. As the weight vector moves away 
from the MMSE weight vector, Wo, the output SNR decreases from the maximum output 
SNROmax = NSNRi ( = 20 for this scenario). As the MSE increases the output SNR 
decreases. 

For a single user, two-weight array the relationship between SNRo and the weights is 
easily shown through the relationship: 

n»rn i w^Rw* w^MLM^w* v^Lv* _ _ 

SNRo+i= 3 ^ = — ^^ = -Y^r- (A-1) 

(T,W w* a„w W* CT.V V* 



113 



114 



Where L and M are the eigenvalue and eigenvector matrices of R respectively. The 



Constant MSE.SNR Contours 



0.5 






-0.5 



1 


/ /sNRo=17.5 


SNRo=14.o/ 


/^V 




/ ^\>^0P^\\ 




SNRo=17^5,^-'-^^'^^ \ 


1^—^-3:^==^ 


SNRo=14.0 


IZ^^^^^/^ 


SNRi=10 


/ 


DOA=0 deg 



-0.5 



0.5 



W(1) 



Figure A. 1 Output contours for a two-element array with a single user. The straight lines 
are constant contours of output SNR. The ellipses are constant MSE contours. 



equation for the constant-value contour is v°Lv - a\{SNRo + l)v^v = . 

For a single user in AWGN the eigenvalues are X,i = On, 'ki = P^ + d^. and the two 
weights in the rotated coordinate system are related by the expression: 



V, = ±v. 



SNRo 



SNRi-SNRo 



(A.2) 



which is a set of lines in two-dimensional, rotated weight-space. The SNR surface in 
weight-space is concave. These relationships have been attributed to Winkler,'^* but the 
original reference does not seem to be available. In the two-dimensional system shown in 



lis 

the figure above the major axis of the constant MSE ellipse is parallel to the noise-only 

eigenvector and the ellipse minor axis is parallel to the signal and noise eigenvector. The 

optimum MMSE weight vector Wo will be almost orthogonal to the noise-only 

eigenvector. 

The gradient of the output SNR is given by: 

„ „.,_ 2SNRol , J SNRo.l 

Vy,SNRo = „ „, u u' 1 w (A.3) 

||wf L SNRi J 

where u is the phase-shift vector defined in equation (1.4) and I is the identity matrix. It 
might be argued that the norm of the gradient of SNRo is a measure of the local variations 
of the SNRo function in weight-space. The Laplacian is the norm-squared of the gradient: 

Vy,SNRof = ^^T^ [NSNRi - SNRo] (A.4) 

||w| 

Some things to note here are that the norm of the gradient is zero when the output SNR is 
maximum ( = NSNRi) and that the gradient norm decreases as the weight vector norm 
increases. It is possible to express output SNR in weight-space angles: 

I T I 

SNRo = SNRi i^]^-J^ = N ■ SNRi cos' a (A.5) 

l|w|| 

where a is the angle between w and u in weight space. If equation (A.5) is substituted into 
equation (A.4) the norm-squared of the gradient becomes: 



W^SNRo 



1 dSNRo^^ 



W da 



(A.6) 



This relationship is apparent in Figure (A. 1) above, variations in SNRo occur when the 
weight vector is displaced in angle a from the phase-shift vector (which is collinear with 
Wo in the figure). 



116 

For a two-user system with a weak user (1) and a strong user (2) the optimum weight 
is almost collinear with the weaker user's phase-shift vector. Perturbations from the 
steady-state weight vector, caused by the noisy gradient estimates of the LMS algorithm, 
will have little effect on the weaker user since Oi (the angle between ui and Wo) is small. 
Or, put another way, the difference term in the brackets of equation ( A.4) is small since 
SNRo] = NSNRij, and the resulting variations are small. For a strong user, the difference in 
the bracketed term may be large since SNR02 ~ SNRoi and the weight variations may cause 
the large fluctuations in the strong user's output. 



10 



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Institute of Brooklyn, 1971 



BIOGRAPHICAL SKETCH 

John Miller received his Bachelor of Science degree from the University of Rorida in 
December, 1981. From February 1982 to April 1989 he worked at the Portable Products 
Division of Motorola Inc. in Plantation, Florida. He returned to the University of Florida 
in August 1989 and was awarded a Masters of Science degree in electrical engineering in 
May 1991. Since this time he has been working towards a Ph.D. 



129 



I certify that I have read this study and that in my opinion it conforms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 




Scott L. Miller, Chairman 
Associate Professor of Electrical 
and Computer Engineering 



I certify that I have read this study and that in my opinion it conforms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosopl 




Le6n W. Couch, II 
Professor of Electrical and Computer 
Engineering 

I certify that I have read this study and that in my opinion it conforms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quaUty, 
as a dissertation for the degree of Doctor of Philosophy. 

John M. M. Anderson 
Assistant Professor of Electrical 
and Computer Engineering 

I certify that I have read this study and that in my opinion it conforms to 
acceptable standards of scholarly presentation and is fuUyjdg^uate, in scope and quality, 
as a dissertation for the degree of Doctor of PhilosopfiyT 



JoseyPnnc _ 

ProressQP^if Eleemcal and Computer 



ingineenn| 

I certify that I have read this study and that in my opinion it conforms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 

Ramesh Shrestha 
Associate Professor of Civil 
Engineering 



This dissertation was submitted to the Graduate Faculty of the College of 
Engineering and to the Graduate School and was accepted as partial fulfillment of the 
requirements for the degree of Doctor of Philosophy. 

August, 1996 r 

/- Winfted M. PhilUps 

Dean, College of Engineering 



Karen A. Holbrook 
Dean, Graduate School 



LD 
1780 

1996 



SCIENCE 
LIBRAJ«Y 



UNIVERSITY OF FLORIDA 



3 1262 08555 0654